Journal of Inequalities in Pure and
Applied Mathematics
ON THE HEISENBERG-PAULI-WEYL INEQUALITY
JOHN MICHAEL RASSIAS
Pedagogical Department, E. E.,
National and Capodistrian University of Athens,
Section of Mathematics and Informatics,
4, Agamemnonos Str., Aghia Paraskevi,
Athens 15342, Greece.
EMail: jrassias@primedu.uoa.gr
volume 5, issue 1, article 4,
2004.
Received 02 January, 2003;
accepted 04 March, 2003.
Communicated by: A. Babenko
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Abstract
In 1927, W. Heisenberg demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom.The
following result named, Heisenberg inequality, is not actually due to Heisenberg. In 1928, according to H. Weyl this result is due to W. Pauli.The said
inequality states, as follows: Assume thatf : R → C is a complex valued function of a random real variable x such that f ∈ L2 (R). Then the product of the
second moment of the random real x for |f|2 and the second moment of the
2
random real ξ for fˆ is at least E|f|2 /4π , where fˆ is the Fourier transform of f,
R
R
√
such that fˆ (ξ) = R e−2iπξx f (x) dx and f (x) = R e2iπξx fˆ (ξ) dξ, i = −1 and
R
E|f|2 = R |f (x)|2 dx. In this paper we generalize the afore-mentioned result
to the higher moments for L2 functions f and establish the Heisenberg-PauliWeyl inequality.
2000 Mathematics Subject Classification: Primary: 26Xxx; Secondary: 42Xxx,
60Xxx, 62Xxx.
Key words: Pascal Identity, Plancherel-Parseval-Rayleigh Identity, Lagrange Identity,
Gaussian function, Fourier transform, Moment, Bessel equation, Hermite polynomials, Heisenberg-Pauli-Weyl Inequality.
We thank the computer scientist C. Markata for her help toward the computation of
the data of Table 1 for the last twenty four cases p = 10, 11, . . ., 33.
We greatly thank Prof. Alexander G. Babenko for the communication and the anonymous referee for his/her comments.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1
Second Order Moment Heisenberg Inequality ([3, 6]) 5
2
Pascal Type Combinatorial Identity . . . . . . . . . . . . . . . . . . . . . 11
3
Generalized Differential Identity . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1
Special cases of (**) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2
Applications of the Recursive Sequence (R) . . . . . . . . 22
3.3
Generalization of the Identity (**) . . . . . . . . . . . . . . . . 23
4
Generalized Plancherel-Parseval-Rayleigh Identity . . . . . . . . . 26
5
The pth -Derivative of the Product of Two Functions . . . . . . . . 30
6
Generalized Integral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.1
Special Cases of (6.2): . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7
Lagrange Type Differential Identity . . . . . . . . . . . . . . . . . . . . . 38
7.1
Special cases of (7.1): . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8
On the Heisenberg-Pauli-Weyl Inequality . . . . . . . . . . . . . . . . 44
8.1
Fourth Order Moment Heisenberg Inequality . . . . . . . 58
8.2
Sixth Order Moment Heisenberg Inequality . . . . . . . . 67
8.3
Eighth Order Moment Heisenberg Inequality . . . . . . . 74
8.4
First Four Generalized Weighted Moment Inequalities 76
8.5
First form of (8.3), if w = 1, xm = 0 and ξm = 0 . . 84
8.6
Second form of (8.3), if ξm = 0 . . . . . . . . . . . . . . . . . 90
8.7
Third form of (8.3), if w = 1 . . . . . . . . . . . . . . . . . . . . 94
9
Gaussian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
9.1
First nine cases of (9.33) . . . . . . . . . . . . . . . . . . . . . . . . 109
9.2
First nine cases of (9.35) . . . . . . . . . . . . . . . . . . . . . . . . 116
9.3
Two Special Cases of (9.56) . . . . . . . . . . . . . . . . . . . . . 134
References
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1.
Introduction
In 1927, W. Heisenberg [9] demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom. In
1933, according to N. Wiener [22] a pair of transforms cannot both be very
small. This uncertainty principle was stated in 1925 by Wiener, according to
Wiener’s autobiography [23, p. 105–107], in a lecture in Göttingen. In 1992,
J.A. Wolf [24] and in 1997, G. Battle [1] established uncertainty principles for
Gelfand pairs and wavelet states, respectively. In 1997, according to Folland
et al. [6], and in 2001, according to Shimeno [14] the uncertainty principle in
harmonic analysis says: A nonzero function and its Fourier transform cannot
both be sharply localized. The following result of the Heisenberg Inequality
is credited to W. Pauli according to H. Weyl [20, p.77, p. 393–394]. In 1928,
according to Pauli [20], the following proposition
holds: the less the uncertainty
2
ˆ
2
in |f | , the greater the uncertainty in f is , and conversely. This result does
not actually appear in Heisenberg’s seminal paper [9] (in 1927). According to
G.B. Folland et al. [6] (in 1997) Heisenberg [9] gave an incisive analysis of the
physics of the uncertainty principle but contains little mathematical precision.
The following Heisenberg inequality provides a precise quantitative formulation
of the above-mentioned uncertainty principle according to Pauli [20].
In what follows we will use the following notation to denote the Fourier
transform of f (x) :
F (f (x)) ≡ [f (x)] ˆ (ξ) .
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1.1.
Second Order Moment Heisenberg Inequality ([3, 6])
For any f ∈ L2 (R), f : R → C, such that
Z
2
|f (x)|2 dx = E|f |2 ,
kf k2 =
R
any fixed but arbitrary constants xm , ξm ∈ R, and for the second order moments
(variances)
Z
(x − xm )2 |f (x)|2 dx
2
(µ2 )|f |2 = σ|f
=
|2
R
and
(µ2 ) fˆ 2 =
| |
σ2ˆ 2
|f |
Z
2
(ξ − ξm ) fˆ (ξ) dξ,
=
R
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E|f2 |2
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(µ2 )|f |2 · (µ2 ) fˆ 2 ≥
,
| |
16π 2
holds, where
fˆ (ξ) =
Z
Z
f (x) =
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e−2iπξx f (x) dx
R
and
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2
the second order moment Heisenberg inequality
(H1 )
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R
Equality holds in (H1 ) iff (if and only if) the Gaussians
2
f (x) = c0 e2πixξm e−c(x−xm ) = c0 e−cx
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2 +2(cx
2
m +iπξm )x−cxm
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hold for some constants c0 ∈ C and c > 0. We note that if xm 6= 0 and ξm = 0,
2
then f (x) = c0 e−c(x−xm ) , c0 ∈ C and c > 0.
Proof. Let xm = ξm = 0, and that the integrals in the inequality (H1 ) be finite.
d
Besides we consider both the ordinary derivative dx
|f |2 = 2 Re f f¯0 and the
Fourier differentiation formula
F f 0 (ξ) = [f 0 (x)] ˆ (ξ) = 2πiξ fˆ (ξ) .
Then we get that the finiteness of the integral
Z
Z 1
ˆ 2
2
0
|f (x)| dx
ξ f (ξ) dξ = 2
4π R
R
implies f 0 ∈ L2 . Integration by parts yields
Z a+
Z a+
d
0
2
x Re f (x) f (x) dx =
x |f (x)|2 dx
dx
a−
a−
a+ Z a+
2
= x |f (x)| −
|f (x)|2 dx,
a−
a−
for −∞ < a− < a+ < ∞. Since f , xf , f 0 ∈ L2 , the integrals in this
equality are finite as a− → −∞ or a+ → ∞ and thus both limits L− =
lim a− |f (a− )|2 and L+ = lim a+ |f (a+ )|2 are finite. These two limits
a− →−∞
a+→∞
L± are equal to zero, for otherwise |f (x)|2 would behave as x1 for big x meaning that f ∈
/ L2 , leading to contradiction. Therefore for the variances about the
origin
Z
x2 |f (x)|2 dx
(m2 )|f |2 = s2|f |2 =
R
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and
(m2 ) fˆ 2 =
| |
s2 ˆ 2
|f |
Z
=
R
2
ξ 2 fˆ (ξ) dξ,
one gets
1
kf k42 =
16π 2
=
=
=
=
≤
≤
2
1
E 2
4π |f |
Z
2
1
2
− |f (x)| dx
16π 2
R
Z
2
1
d
2
x |f (x)| dx
16π 2 R dx
Z
2
1
2 x Re f (x) f 0 (x) dx
16π 2
R
Z 2
1 1
0
0
xf (x) f (x) + xf (x)f (x) dx
4π 2 2 R
Z
2
1
0
|xf (x) f (x)| dx
4π 2 R
Z
Z
1
2
2
2
0
x |f (x)| dx
|f (x)| dx = s2|f |2 · s2 ˆ 2 .
|f |
4π 2
R
R
Equality in these inequalities holds iff the differential equation f 0 (x) = −2cxf (x)
2
of first order holds for c > 0 or if the Gaussians f (x) = c0 e−cx hold for some
constants c0 ∈ C, and c > 0.
Assuming any fixed but arbitrary real constants xm , ξm and employing the
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transformation
fxm ,ξm (x) = e2πixξm f (x − xm ) ,
xδ = x − xm 6= 0,
we establish the formula
fˆxm ,ξm (ξ) = e−2πixm (ξ−ξm ) fˆ (ξ − ξm ) = e2πixm ξm fˆξm ,−xm (ξ) .
Therefore the map f → fxm ,ξm preserves all L2p (p ∈ N) norms of f and fˆ
while shifting the centers of mass of f and fˆ by real xm and ξm , respectively.
Therefore equality holds in (H1 ) for any fixed but arbitrary constants xm , ξm ∈
R iff the general formula
f (x) = c0 e2πixξm e−c(x−xm )
dfa
dfa dxδ
dfa
=
=
=
,
dx
dxδ dx
dxδ
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In fact,
ln |fa | = −c (x − xm )2 ,
or
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2
holds for some constants c0 ∈ C, and c > 0. One can observe that this general
formula is the complete (general) solution of the following a-differential equation fa0 = −2c (x − xm ) fa , by the method of the separation of variables, where
a = −2πξm i, fa = eax f . We note that
fa0
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2
eax f (x) = fa (x) = c0 e−c(x−xm ) ,
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or
2
f (x) = c0 e−ax e−c(x−xm ) .
This is a special case for the equality of the general formula (8.3) of our theorem in Section 8, on the generalized weighted moment Heisenberg uncertainty
principle. Therefore the proof of this fundamental Heisenberg Inequality (H1 )
is complete.
2
2
We note that, if f ∈ L (R) and the L -norm of f is kf k2 = 1 = fˆ , then
2
2
ˆ
2
|f | and f are both probability density functions. The Heisenberg inequality
in mathematical statistics and Fourier analysis asserts that: The product of the
2
variances of the probability measures |f (x)|2 dx and fˆ (ξ) dξ is larger than
an absolute constant. Parts of harmonic analysis on Euclidean spaces can naturally be expressed in terms of a Gaussian measure; that is, a measure of the
2
form c0 e−c|x| dx, where dx is the Lebesgue measure and c, c0 (> 0) constants.
Among these are: Logarithmic Sobolev inequalities, and Hermite expansions.
Finally one [14] observes that:
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1
2
σ|f
· σ 2 ˆ 2 ≥ kf k42 ,
|2
|f |
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1
fˆ (ξ) = √
2π
Z
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1
f (x) = √
2π
Z
if f ∈ L2 (R),
and
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Inequality
e−iξx f (x) dx
R
R
eiξx fˆ (ξ) dξ,
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where the L2 -norm kf k2 is defined as in (H1 ) above.
In 1999, according to Gasquet et al. [8] the Heisenberg inequality in spectral
analysis says that the product of the effective duration ∆x and the effective
1
∗
bandwidth ∆ξ of a signal cannot be
. less than the value 4 π = H (=Heisenberg
2
lower bound), where ∆x2 = σ|f
|2
∆ξ
2
=
σ2ˆ 2
|f |
E|f |2 and
E fˆ 2
| |
= σ2ˆ 2
|f |
.
E|f |2
with f : R → C, fˆ : R → C defined as in (H1 ), and
Z ˆ 2
E|f |2 =
f (ξ) dξ = E|fˆ|2 .
R
In this paper we generalize the Heisenberg inequality to the higher moments for
L2 functions f and establish the Heisenberg-Pauli-Weyl inequality.
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2.
Pascal Type Combinatorial Identity
We state and prove the new Pascal type combinatorial identity.
Proposition 2.1. If 0 ≤ k2 is the greatest integer ≤ k2 , then
k
k−1 k−i
k+1
k−i
k−i+1
(C)
+
=
,
i
i−1
i
k−i
k−i
k−i+1
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, and 0 ≤ i ≤ k2 for
k
i ∈ N0 = {0, 1, 2, . . .} such that −1
= 0.
Note that the classical Pascal identity is
k−i
k−i
k−i+1
+
=
,
i
i−1
i
k
.
0≤i≤
2
Proof. It is clear that k (k − 2i + 1) + (k − 1) i = (k − i) (k + 1). Thus
k
k−1 k−i
k−i
+
i
i−1
k−i
k−i
k
(k − i)!
k−1
(k − i)!
=
+
k − i i! (k − 2i)! k − i (i − 1)! (k − 2i + 1)!
(k−i+1)!
(k−i+1)!
k
k−1
k−i+1
k−i+1
=
+
(k−2i+1)!
k − i i!
k − i i!i (k − 2i + 1)!
k−2i+1
(k − i + 1)!
1
=
[k (k − 2i + 1) + (k − 1) i]
i! (k − 2i + 1)! (k − i) (k − i + 1)
k+1
k−i+1
,
=
i
k−i+1
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completing the proof of this identity.
Note that all of the three combinations: k−i
,
i
k
are positive numbers if 1 ≤ i ≤ 2 for i ∈ N.
k−i
i−1
, and
k−i+1
i
exist and
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3.
Generalized Differential Identity
We state and prove the new differential identity.
Proposition
function of a real variable
3.1. If f : R → C is a complex valued
dj
x, 0 ≤ k2 is the greatest integer ≤ k2 , f (j) = dx
f
,
and (·) is the conjugate of
j
(·), then
(*)
f (x) f (k) (x) + f (k) (x) f¯ (x)
=
[ k2 ]
X
i=0
(−1)i
k
k−i
k−i
i
On the Heisenberg-Pauli-Weyl
Inequality
2
dk−2i (i)
,
f
(x)
dxk−2i
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤
i ∈ N0 = {0, 1, 2, . . .}.
Note that for k = 1 we have
f (x) f (1) (x) + f (1) (x) f¯ (x)
1−2·0
(0)
1
1
d
1−0
(=0)
[
]
f (x)2
2
= (−1)
0
1−0
dx1−2·0
d
=
|f (x)|2
dx
(1)
= |f (x)|2
.
k
2
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If we denote Gk (f ) = f f (k) + f (k) f¯, and
2 (k−2i)
dk−2i 2
f (i) = k−2i f (i) dx
2
dk−2i di = k−2i i f ,
dx
dx
k
0≤i≤
for k ∈ N = {1, 2, . . .}
2
On the Heisenberg-Pauli-Weyl
Inequality
and i ∈ N0 , then (*) is equivalent to
(**)
Gk (f ) =
[ k2 ]
X
i=0
k
(−1)
k−i
i
k−i
i
(i) 2 (k−2i)
f .
Proof. For k = 1 (**) is trivial. Assume that (**) holds for k and claim that it
holds for k + 1. In fact,
(1)
(1)
Gk (f ) = f f (k) + f (k) f¯
(1)
(1)
+ f (k) f¯
= f f (k)
= f (1) f (k) + f f (k+1) + f (k+1) f¯ + f (k) f (1)
(1) (k)
(k) (1)
(k+1)
(k+1) ¯
+ ff
+f
f
= f f +f f
(k−1) (1)
(k−1)
= f (1) (f (1) )
+ Gk+1 (f )
+ f (1)
f
= Gk−1 f (1) + Gk+1 (f ) ,
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or the recursive sequence
(1)
Gk+1 (f ) = Gk (f ) − Gk−1 f (1) ,
(R)
2
(1)
for k ∈ N = {1, 2, . . .}, with G0 f (1) = f (1) , and G1 (f ) = |f |2 .
From the induction hypothesis, the recursive relation (R), the fact that
j
X
λi+1 =
i=0
j+1
X
λi , (−1)i−1 = − (−1)i
i=1
for i ∈ N0 , and
John Michael Rassias
 k
 2 − 1,
k−1
=
 k
2
2
such that
k
2
≤
On the Heisenberg-Pauli-Weyl
Inequality
k−1 2
=
k = 2v for v = 1, 2, . . .
k−1
2
, k = 2λ + 1 for λ = 0, 1, . . .
+ 1, if k ∈ N, we find
Gk+1 (f )


k


[
]
2
X

k
k − i (i) 2 (k+1−2i)
i
=
(−1)
f
i


k−i
 i=0



k−1


2 ]
[X
(i+1) 2 (k−1−2i) 
k−1
k−1−i
i
f
−
(−1)
i


k−1−i
 i=0

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= (−1)0
+
[ k2 ]
X
i=1
k
k−0
(−1)i
k
k−i
k−0
0
k−i
i
(0) 2 (k+1−2·0)
f 

(i) 2 (k+1−2i) 
f 



k−1
+1


[
]
2
 X

(−1)i−1 (k − 1)
k − 1 − (i − 1) (i) 2 (k−1−2(i−1))
−
f
i−1



 i=1 (k − 1) − (i − 1)


k


[X
2]

(i) 2 (k+1−2i) 
k
k−i
2 (k+1)
i
=
|f |
+
f
(−1)
i


k−i


i=1


k−1
+1


2 ]

[ X
k − i (i) 2 (k+1−2i)
i k−1
f
+
(−1)
i−1


k−i

 i=1


k


[
]
2


X
k
k − i (i) 2 (k+1−2i)
2 (k+1)
i
=
|f |
+
(−1)
f
i


k−i


i=1




[ k2 ]
X

(i) 2 (k+1−2i)
k−i
i k−1
+
(−1)
f
+ S (f ) ,
i−1


k−i
 i=1

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where

0,
if k = 2v for v = 1, 2, . . .







k−1 
+1
[ k−1

2 ]

 (−1) (k −1) k− 2 + 1
k−1
+1 −1
k − k−1
+1
S (f ) =
2
2


+1))

2 (k+1−2([ k−1
k−1
2 ]


+1
]
)
([

2
× f
,





if k = 2λ + 1 for λ = 0, 1, . . . .
John Michael Rassias
Thus
[ k2 ]
X
k
(k+1)
k−i
i
2
Gk+1 (f ) = |f |
(−1)
+
i
k−i
i=1
(i) 2 (k+1−2i)
k−1 k−i
f +
+ S (f ) ,
i−1
k−i
where
S (f ) =
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
0,







(−1)[







if k = 2v for v = 1, 2, . . .
k−1
2
]
1−k
k−1−[ k−1
2 ]
2 (k−1−2[ k−1
2 ])
k−1
k − 1 − k−1
+1
([
]
)
2
2
f
,
k−1
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2
if k = 2λ + 1 for λ = 0, 1, . . .
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=

0,



if k = 2v for v = 1, 2, . . .
k−1


 (−1) 2
=
1−k
k−1
2
k−1
2
k−1
2
2 (0)
( k+1
)
2
, if k = 2λ + 1 for λ = 0, 1, . . .
f

0,



if k = 2v for v = 1, 2, . . .
k+1


 2 (−1) 2
2 ( k+1
)
2
f
, if k = 2λ + 1 for λ = 0, 1, . . .
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
=
because

0,







if k = 2v for v = 1, 2, . . .
Title Page
k+1
(−1)[ 2 ]







k+1
k+1−[ k+1
2 ]
2 (k+1−2[ k+1
2 ])
k + 1 − k+1
([ k+1
])
2
2
,
f
k+1
2
if k = 2λ + 1 for λ = 0, 1, . . . ,
k+1
k+1
=
, if k = 2λ + 1 for λ = 0, 1, . . .
2
2
In addition
k we note
k+1that,
from the above Pascal type combinatorial identity (C)
k
and 2 = 2 = 2 , if k = 2v for v = 1, 2, . . . , one gets
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k k ) 2 (k+1−2[ k2 ])
k
k
−
1
k
−
k
−
[
]
([
])
2
2
k
(−1)
+
f 2 k
k
−1
k− 2
2
2
k
2 (k+1−2[ k2 ])
k
k
k
+
1
k
−
+
1
[
]
([
])
k2 k
= (−1) 2
f 2 k− 2 +1
2
k+1 2 (k+1−2[ k+1
2 ])
k+1
k+1
k
+
1
k
+
1
−
[
]
([
])
2
k+1 k+1 = (−1) 2
,
f 2 k+1− 2
2
(
k
2
k
k − k2
if k = 2v for v = 1, 2, . . . . From these results, one
obtains
(0) 2 (k+1−2·0)
k+1
k+1−0
f Gk+1 (f ) = (−1)0
0
k+1−2·0
+
[ k2 ]
X
i=1
=
k+1
[X
2 ]
i=1
k+1
(−1)
k+1−i
i
k+1
(−1)
k+1−i
i
k+1−i
i
k+1−i
i
(i) 2 (k+1−2i)
f +S (f )
(i) 2 (k+1−2i)
f ,
completing the proof of (**) for k + 1 and thus by the induction principle on k,
(**) holds for any k ∈ N.
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3.1.
Special cases of (**)
Page 19 of 151
2 (1)
G1 (f ) = |f |
,
2
(2)
G2 (f ) = |f |2
− 2 f (1) ,
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2 (3)
G3 (f ) = |f |
2 (4)
G4 (f ) = |f |
2 (5)
G5 (f ) = |f |
2 (6)
G6 (f ) = |f |
2 (7)
G7 (f ) = |f |
(1)
(1) 2
−3 f
,
(2)
2
(1) 2
−4 f
+ 2 f (2) ,
(3)
(1)
(1) 2
(2) 2
+5 f
,
−5 f
(4)
(2)
2
(1) 2
(2) 2
−6 f
+9 f
− 2 f (3) ,
(5)
(3)
(1)
(1) 2
(2) 2
(3) 2
−7 f
+ 14 f
−7 f
,
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
2
2 (2)
2 (6)
2 (4)
(8)
+2 f (4) .
−16 f (3) G8 (f ) = |f |2 −8 f (1) +20 f (2) Title Page
We note that if one takes the above numerical coefficients of Gi (f ) (i = 1, 2, . . ., 8)
absolutely, then one establishes the pattern with
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
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a1 = 2
a2 = 5
a3 = 9
a4 = 14
a5 = 20
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b1 = 2
b2 = 7
b3 = 16
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a1 = 2
a2 = 2 + 3 = 5
a3 = 2 + 3 + 4 = 9
a4 = 2 + 3 + 4 + 5 = 14
a5 = 2 + 3 + 4 + 5 + 6 = 20
b 1 = a1 = 2
b 2 = a1 + a2 = 7
b3 = a1 + a2 + a3 = 16
c1 = b1 = a1 = 2
Following this pattern we get
2 (9)
G9 (f ) = |f |
(7)
(5)
(1) 2
(2) 2
−9 f
+ a6 f
(3)
(1)
(3) 2
(4) 2
− b4 f
+ c2 f
,
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
Title Page
where
Contents
a6 = a5 + 7 = 27,
b4 = b3 + a4 = 16 + 14 = 30,
c2 = c1 + b2 = 2 + 7 = 9.
Similarly, one gets
G10 (f ) = |f |2
(10)
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2 (8)
2 (6)
+ a7 f (2) − 10 f (1) 2 (4)
2 (2)
2
− b5 f (3) + c3 f (4) − d1 f (5) ,
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where
a7
b5
c3
d1
3.2.
= a6 + 8 = 35,
= b4 + a5 = 30 + 20 = 50,
= c2 + b3 = 9 + 16 = 25,
= c1 = b1 = a1 = 2.
Applications of the Recursive Sequence (R)
(1)
(1)
G4 (f ) = G3 (f ) − G2 f
(1) (2)
(2)
= f f (3) + f (3) f
− f (1) (f (1) ) + f (1)
f (1)
(2)
(1) 2 (2)
(2) 2
2 (4)
(1) 2
−3 f
−2 f
= |f |
−
f
(2)
2
2 (4)
(1) 2
= |f |
−4 f
+ 2 f (2) and
(1)
G5 (f ) = G4 (f ) − G3 f (1)
(3)
(1)
2 (5)
(1) 2
(2) 2
= |f |
−4 f
+2 f
(1) 2 (3)
(1)
(2) 2
−
−3 f
f
= |f |2
(5)
2 (3)
2 (1)
− 5 f (1) + 5 f (2) ,
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yielding also the above generalized differential identity (**) for k = 3 and
k = 4, respectively.
3.3.
Generalization of the Identity (**)
We denote
Hkl (f ) = f (l) f (k) + f (k) f (l) .
It is clear that

(k−l)
(k−l)

+ f (l)
f (l) ,
if k > l
Gk−l f (l) = f (l) (f (l) )





2
Hkl (f ) =
G0 f (l) = 2 f (l) ,
if k = l





(l−k)

(l−k)
+ f (k)
f (k) , if k < l
Gl−k f (k) = f (k) (f (k) )
From these and (**) we conclude that
 k−l
[ 2 ]


 X
k−l
k − l − i (i+l) 2 (k−l−2i)
i


(−1)
f
, if k > l


i
k−l−i


i=0



 2
Hkl (f ) =
2 f (l) ,
if k = l





l−k


[X
2 ]


(i+k) 2 (l−k−2i)
l
−
k
l
−
k
−
i

i

f
, if k < l
(−1)


i
l
−
k
−
i
i=0
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For instance, if k = 3 > 2 = l, then from this formula one gets
H32 (f ) = G3−2 f (2)
3−2
[X
2 ]
3−2
3 − 2 − i (i+2) 2 (3−2−2i)
=
(−1)
f
i
3
−
2
−
i
i=0
1
1 − 0 (2) 2 (1−2·0)
0
f
= (−1)
0
1−0
(1)
(2) 2
= f
.
i
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
In fact,
Title Page
H32 (f ) = f (2) f (3) + f (3) f (2)
(1)
(1)
= f (2) (f (2) ) + f (2)
f (2)
2 (1)
= G1 f (2) = f (2) .
Another special case, if k = 3 > 1 = l, then one gets
H31 (f ) = G3−1 f (1)
=
3−1
[X
2 ]
i=0
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3−1
(−1)i
3−1−i
3−1−i
i
(i+1) 2 (3−1−2i)
f
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2 − 0 (1) 2 (2−2·0)
f
0
2
2 − 1 (2) 2 (2−2·1)
1
+ (−1)
f
1
2−1
2 (2)
2
= f (1) − 2 f (2) .
2
= (−1)
2−0
0
In fact,
H31 (f ) = f (1) f (3) + f (3) f (1)
(2)
(2)
= f (1) (f (1) ) + f (1)
f (1)
2
2 (2)
= G2 f (1) = f (1) − 2 f (2) .
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John Michael Rassias
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4.
Generalized Plancherel-Parseval-Rayleigh Identity
We state and prove the new Plancherel-Parseval-Rayleigh identity.
Proposition 4.1. If f and fa : R → C are complex valued functions of a real
√
(p)
dp
variable x, fa = eax f and fa = dx
−1 and
p fa , where a = −2πξm i with i =
ˆ
any fixed but arbitrary constant ξm ∈ R, f the Fourier transform of f , such that
Z
ˆ
f (ξ) =
e−2iπξx f (x) dx
R
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
with ξ real and
Z
f (x) =
e2iπξx fˆ (ξ) dξ,
R
(p)
fa ,
as well as f ,
and (ξ − ξm ) fˆ are in L2 (R), then
Z
Z
2
(p)
1
2p ˆ
fa (x)2 dx
(4.1)
(ξ − ξm ) f (ξ) dξ =
(2π)2p R
R
Contents
p
holds for any fixed but arbitrary p ∈ N0 = {0, 1, 2, . . .}.
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Proof. Denote
(4.2)
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g (x) = e−2πixξm f (x + xm ) ,
for any fixed but arbitrary constant xm ∈ R.
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From (4.2) one gets that
Z
(4.3)
ĝ (ξ) =
e−2iπξx g (x) dx
ZR
=
e−2iπξx e−2πixξm f (x + xm ) dx
R
Z
2πixm (ξ+ξm )
=e
e−2πi(ξ+ξm )x f (x) dx
R
2πixm (ξ+ξm )
=e
fˆ (ξ + ξm ) .
(p)
(p)
Denote the Fourier
transform
of
g(p)
either
by
F
g
(ξ)
,
or
F
g
(x)
(ξ) , or
also as g (p) (x) ˆ (ξ) .
From this and Gasquet et al. [8, p. 155–157] we find
1
ξ p ĝ (ξ) =
F g (p) (ξ) .
(2πi)p
(4.4)
Also denote
hp (x) = g (p) (x) .
(4.5)
From (4.5) and the classical Plancherel-Parseval-Rayleigh identity one gets
Z Z
2
|hp (x)|2 dx,
ĥp (ξ) dξ =
R
R
(4.6)
R
(p) 2
F g (ξ) dξ =
Z
R
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or
Z
On the Heisenberg-Pauli-Weyl
Inequality
(p)
g (x)2 dx.
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Finally denote
Z
(4.7)
(µ2p ) fˆ 2 =
| |
R
2
(ξ − ξm )2p fˆ (ξ) dξ
2
moment of ξ for fˆ for any fixed but arbitrary constant ξm ∈ R and
the 2pth
p ∈ N0 .
Substituting ξ with ξ+ ξm we find from (4.3) – (4.7) that
Z
2
(µ2p ) fˆ 2 =
ξ 2p fˆ (ξ + ξm ) dξ
| |
ZR
2
2p 2πixm (ξ+ξm ) ˆ
=
ξ e
f (ξ + ξm ) dξ
ZR
=
ξ 2p |ĝ (ξ)|2 dξ
R
Z
(p) 2
1
F g (ξ) dξ
=
2p
(2π)
ZR
(p)
1
g (x)2 dx.
=
2p
(2π)
R
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John Michael Rassias
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From this and (4.2) we find
1
(µ2p ) fˆ 2 =
| |
(2π)2p
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Z (p) 2
−2πixξm
f (x + xm ) dx.
e
Page 28 of 151
R
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Placing x − xm on x in this identity one gets
Z (p) 2
1
−2πi(x−xm )ξm
(µ2p ) fˆ 2 =
e
f
(x)
dx
| |
(2π)2p R
Z 2
1
ax
(p) (e
f
(x))
=
dx.
(2π)2p R
Employing eax f (x) = fa (x) in this new identity we find
Z
(p)
1
fa (x)2 dx,
(µ2p ) fˆ 2 =
| |
(2π)2p R
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
completing the proof of the required identity (4.1) ([16] – [24]).
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5.
The pth-Derivative of the Product of Two Functions
We state and outline a proof for the following well-known result on the pth derivative of the product of two functions.
Proposition 5.1. If fi : R → C (i = 1, 2) are two complex valued functions of
a real variable x, then the pth -derivative of the product f1 f2 is given, in terms
(m)
(p−m)
of the lower derivatives f1 , f2
by
(p)
(f1 f2 )
(5.1)
p X
p
(m) (p−m)
=
f1 f2
m
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
m=0
Title Page
for any fixed but arbitrary p ∈ N0 = {0} ∪ N.
Contents
(0) (0)
Proof. In fact, for p = 0 the formula (5.1) is trivial, as (f1 f2 )(0) = f1 f2 .
(1)
(1)
When p = 1 the formula (5.1) is (f1 f2 )(1) = f1 f2 + f1 f2 which holds.
Assume that (5.1) holds, as well. Differentiating this formula we get
(p+1)
(f1 f2 )
p p X
X
p
p
(m+1) (p−m)
(m) (p+1−m)
=
f1
f2
+
f1 f2
m
m
m=0
m=0
p+1 p X
X
p
p
(m) (p+1−m)
(m) (p+1−m)
=
f1 f2
+
f1 f2
m−1
m
m=1
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=
p
p
(p+1)
f1
f2
+
p X
m=1
+
p
0
p
m−1
+
p
m
(m) (p+1−m)
f1 f2
(p+1)
f1 f2
p X
p+1
(m) (p+1−m)
=
+
f1 f2
m
m=1
p+1
(p+1)
+
f1 f2
0
p+1 X
p+1
(m) (p+1−m)
=
f1 f2
,
m
p+1
p+1
(p+1)
f1
f2
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
m=0
Title Page
as the classical Pascal identity
p
p
p+1
(P)
+
=
m−1
m
m
holds for m ∈ N : 1 < m ≤ p. Therefore by induction on p the proof of (5.1) is
complete.
Employing√the formula (5.1) with f1 (x) = f (x), f2 (x) = eax , where a =
−2πξm i, i = −1, ξm fixed but arbitrary real, and placing fa (x) = (f1 f2 )(x) =
eax f (x), one gets
p X
p
(p)
ax
(5.2)
fa (x) = e
ap−m f (m) (x) .
m
m=0
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Similarly from the formula (5.1) with f1 (x) = f (x), f2 (x) = f (x), |f |2 = f f
and p = k, m = j, we get the following formula
k X
k
2 (k)
(5.3)
|f |
=
f (j) f (k−j) ,
j
j=0
for the k th derivative of |f |2 .
(p)
Note that from (5.2) with m = k one gets the modulus of fa to be of the
form
p (p) X
p
fa = (5.4)
ap−k f (k) ,
k
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
k=0
because |eax | = 1 by the Euler formula : eiθ = cos θ+i sin θ, with θ = −2πξm x
(∈ R).
Also note that the 2pth moment of the real x for |f |2 is of the form
Z
(5.5)
(µ2p )|f |2 =
(x − xm )2p |f (x)|2 dx
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for any fixed but arbitrary constant xm ∈ R and p ∈ N0 .
Placing x + xm on x in (5.5) we find
Z
(µ2p )|f |2 =
x2p |f (x + xm )|2 dx.
R
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From this, (4.2) and
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|g (x)| = e−2πixξm f (x + xm ) = |f (x + xm )|
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one gets that
Z
(5.6)
(µ2p )|f |2 =
x2p |g (x)|2 dx.
R
Besides (5.5) and |fa | = |f | yield
Z
(5.7)
(µ2p )|f |2 =
(x − xm )2p |fa (x)|2 dx.
R
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John Michael Rassias
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6.
Generalized Integral Identities
We state and outline a proof for the following well-known result on integral
identities.
Proposition 6.1. If f : R → C is a complex valued function of a real variable
x, and h : R → R is a real valued function of x, as well as, w, wp : R → R are
two real valued functions of x, such that wp (x) = (x − xm )p w(x) for any fixed
but arbitrary constant xm ∈ R and v = p − 2q, 0 ≤ q ≤ p2 , then
On the Heisenberg-Pauli-Weyl
Inequality
i)
Z
(6.1)
wp (x) h(v) (x) dx
=
v−1
X
r
(−1)
wp(r)
John Michael Rassias
(v−r−1)
(x) h
v
(x) + (−1)
Z
wp(v) (x) h (x) dx
Contents
r=0
holds for any fixed but arbitrary p ∈ N0 = {0} ∪ N and v ∈ N, and all
r : r = 0, 1, 2, . . ., v − 1, as well as
ii)
Z
(v)
wp (x) h
(6.2)
v
Z
(x) dx = (−1)
R
R
v−1
X
r=0
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wp(v) (x) h (x) dx
holds if the condition
(6.3)
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(−1)r lim wp(r) (x) h(v−r−1) (x) = 0,
Page 34 of 151
|x|→∞
holds, and if all these integrals exist.
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Proof. The proof is as follows.
i) For v = 1 the identity (6.1) holds, because by integration by parts one gets
Z
Z
Z
(1)
wp h
(x) dx = wp (x) dh (x) = (wp h) (x) −
wp(1) h (x) dx.
Assume that (6.1) holds for v. Claim that (6.1) holds for v + 1, as well. In
fact, by integration by parts and from (6.1) we get
Z
wp h(v+1) (x) dx
Z
= wp (x) dh(v) (x)
Z
(v)
= wp h
(x) − wp(1) (x) h(v) (x) dx
!
" v−1
X
r
= wp h(v) (x) −
(−1) wp(r+1) h(v−r−1) (x)
r=0
v
Z
+ (−1)
wp(v+1) h
(x) dx
" v
#
X
= wp h(v) (x) −
(−1)r−1 wp(r) h(v−(r−1)−1) (x)
r=1
v
− (−1)
Z
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John Michael Rassias
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wp(v+1) h
(x) dx
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v
X
= wp h(v) (x) +
!
r
(−1) wp(r) h((v+1)−r−1)
(x)
r=1
v+1
+ (−1)

=
(v+1)−1
X
r
(−1)
Z
wp(v+1) h (x) dx

wp(r) h((v+1)−r−1)  (x)
v+1
+ (−1)
Z
wp(v+1) h (x) dx,
r=0
which, by induction principle on v, completes the proof of the integral
identity (6.1).
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
ii) The proof of (6.2) is clear from (6.1) and (6.3).
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6.1.
Special Cases of (6.2):
2
i) If h (x) = f (l) (x) and v = p − 2q, then from (6.2) one gets
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Z
(6.4)
2 (p−2q)
wp (x) f (l) (x)
dx
R
Z
2
p−2q
= (−1)
wp(p−2q) (x) f (l) (x) dx,
R
for 0 ≤ l ≤ q ≤
p
,
2
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ii) If
h (x) = Re rqkj f (k) (x) f (j) (x) ,
and if (6.3) holds, then from (6.2) we get
Z
(6.5)
(p−2q)
dx
wp (x) Re rqkj f (k) (x) f (j) (x)
R
Z
p−2q
(p−2q)
(k)
(j)
= (−1)
wp
(x) Re rqkj f (x) f (x) dx,
R
where rqkj = (−1)q−
k+j
2
for 0 ≤ k < j ≤ q ≤
p
, and v = p − 2q.
2
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Inequality
John Michael Rassias
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7.
Lagrange Type Differential Identity
We state and prove the new Lagrange type differential identity.
Proposition 7.1. If f : R → C is a complex √
valued function of a real variable
x, and fa = eax f , where a = −βi, with i = −1 and β = 2πξm for any fixed
but arbitrary real constant ξm , as well as if
2
p
Apk =
β 2(p−k) ,
0 ≤ k ≤ p,
k
and
Bpkj = spk
p
k
p
j
β 2p−j−k ,
0 ≤ k < j ≤ p,
where spk = (−1)p−k (0 ≤ k ≤ p), then
X
(p) 2 X
2
fa =
Apk f (k) + 2
Bpkj Re rpkj f (k) f (j) ,
k=0
0≤k<j≤p
holds for any fixed but arbitrary p ∈ N0 = {0} ∪ N, where (·) is the conjugate
k+j
of (·), and rpkj = (−1)p− 2 (0 ≤ k < j ≤ p), and Re (·) is the real part of
(·).
Note that spk ∈ {±1} and rpkj ∈ {±1, ±i}.
Proof. In fact, the classical Lagrange identity
p
2
!
!
p
p
X
X
X
(7.2)
rk zk =
|rk |2
|zk |2 −
k=0
k=0
k=0
John Michael Rassias
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p
(7.1)
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X
0≤k<j≤p
|rk zj − rj zk |2 ,
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with rk , zk ∈ C, such that 0 ≤ k ≤ p, takes the new form
2 " p
p
!
!
p
X
X
X
X
rk zk =
|rl |2
|zk |2 −
|rk |2 |zj |2
k=0
l=0
k=0
0≤k<j≤p
#
X
X
−
|rj |2 |zk |2 + 2
Re (rk rj zk zj )
0≤k<j≤p
"
=
0≤k<j≤p
p
p
X
X
|rl |2 −
l=0
+ ··· +
p
X
!
|rk |2
|z0 |2 +
k6=0
l=0
p
p−1
X
X
|rl |2 −
|rl |2 −
!
|rk |2
!
|rk |2
|z1 |2
|zp |2 + 2
John Michael Rassias
X
Re (rk rj zk zj ),
0≤k<j≤p
or the new identity
p
2
p
X
X
X
(7.3)
rk zk =
|rk |2 |zk |2 + 2
Re (rk rj zk zj ),
k=0
k=0
0≤k<j≤p
because
(7.4)
On the Heisenberg-Pauli-Weyl
Inequality
k6=1
#
k6=p
l=0
p
X
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|rk zj − rj zk |2 = (rk zj − rj zk ) (rk zj − rj zk )
= |rk |2 |zj |2 + |rj |2 |zk |2 − (rk rj zk zj + rk rj zk zj )
2
2
2
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2
= |rk | |zj | + |rj | |zk | − 2 Re (rk rj zk zj ) ,
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as well as
X
(7.5)
|rk |2 |zj |2 +
|rj |2 |zk |2
0≤k<j≤p
0≤k<j≤p
=
X
X
2
2
|rk | |zj |
0≤k6=j≤p
!
=
X
2
!
2
|z0 | +
|rk |
X
k6=0
2
|rk |
!
2
|z1 | + · · · +
k6=1
X
2
|rk |
|zp |2 .
k6=p
On the Heisenberg-Pauli-Weyl
Inequality
Setting
John Michael Rassias
(7.6)
rk =
p
k
ap−k =
p
k
p
k
2
p
j
(−βi)p−k = (−i)p−k
p
k
one gets that
β p−k ,
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|rk |2 =
(7.7)
β 2(p−k) = Apk ,
and
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(7.8)
p
k
(−βi)p−k (βi)p−j
p
p
p−k
2p−k−j
=i
(−1)
β 2p−k−j
k
j
p
p
= rpkj spk
β 2p−k−j = Bpkj rpkj ,
k
j
r k rj =
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where Apk , Bpkj ∈ R, and spk = (−1)p−k ∈ {±1} as well as
rpkj = i2p−k−j = (−1)p−
k+j
2
Thus employing (5.4) and substituting
p
(k)
zk = f , rk =
ap−k
k
∈ {±1, ±i} .
(0 ≤ k ≤ p),
in (7.3), we complete from (7.6) – (7.8), the proof of the identity (7.1).
We note that
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
spk = 1,
if p ≡ k(mod 2);
spk = −1, if p ≡ (k + 1) mod 2.
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Similarly we have
rpkj = 1,
if 2p ≡ (k + j)(mod 4);
rpkj = i,
if 2p ≡ (k + j + 1)(mod 4);
rpkj = −1, if 2p ≡ (k + j + 2)(mod 4);
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rpkj = −i, if 2p ≡ (k + j + 3)(mod 4).
Finally (7.2) may be called the Lagrange identity of first form, and (7.3) the
Lagrange identity of second form.
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7.1.
Special cases of (7.1):
(i)
(7.9)
(1) 2 ax 0 2
fa = (e f ) 2
= A10 |f |2 + A11 |f 0 | + 2B101 Re r101 f f 0
2
= β 2 |f |2 + |f 0 | + 2 (−β) Re if f 0
2
= β 2 |f |2 + |f 0 | + 2β Im f f 0 ,
because
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
(7.10)
Re (iz) = − Im (z) ,
where Im (z) is the imaginary part of z ∈ C.
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Also note that another way to find (7.9), is to employ directly only (5.4),
as follows:
(1) 2
fa = |af + f 0 |2
Contents
2
= |−iβf + f 0 |
= (−iβf + f 0 ) iβf + f 0
2
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= β 2 |f |2 + |f 0 | + β (−i) f f 0 − f f 0
2
= β 2 |f |2 + |f 0 | + 2β Im f f 0 .
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(ii)
(7.11)
(2) 2 ax 00 2
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2
2
= A20 |f |2 + A21 |f 0 | + A22 |f 00 |
+ 2 Re r201 f f 0 + r202 f f 00 + r212 f 0 f 00
2
2
= β 4 |f |2 + 4β 2 |f 0 | + |f 00 |
+ 2 Re −2iβ 3 f f 0 − β 2 f f 00 − 2iβf 0 f 00
2
2
= β 4 |f |2 + 4β 2 |f 0 | + |f 00 | + 4β 3 Im f f 0
− 2β 2 Re f f 00 + 4β Im f 0 f 00 .
Similarly, from (5.4) we get also (7.11), as follows:
(2) 2 2
fa = a f + af 0 + f 00 2 = −β 2 f − iβf 0 + f 00 −β 2 f + iβf 0 + f 00 ,
leading easily to (7.11).
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Inequality
John Michael Rassias
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8.
On the Heisenberg-Pauli-Weyl Inequality
We assume that f : R → C is a complex valued function of a real variable x, and
w : R → R a real valued weight function of x, as well as xm , ξm any fixed
√ but
arbitrary real constants. Denote fa = eax f , where a = −2πξm i with i = −1,
and fˆ the Fourier transform of f , such that
Z
ˆ
f (ξ) =
e−2iπξx f (x) dx
R
and
Z
f (x) =
On the Heisenberg-Pauli-Weyl
Inequality
e2iπξx fˆ (ξ) dξ.
John Michael Rassias
R
Also we denote
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Z
(µ2p )w,|f |2 =
Contents
w2 (x) (x − xm )2p |f (x)|2 dx
R
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J
2
the 2pth weighted moment of x for |f | with weight function w and
Z
2
2p ˆ
(µ2p ) fˆ 2 =
(ξ − ξm ) f (ξ) dξ
| |
R
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2
th
the 2p moment of ξ for fˆ . Besides we denote
Cq = (−1)q
p
p−q
p−q
q
, if 0 ≤ q ≤
hpi 2
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= the greatest integer ≤
p
2
,
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p−2q
Z
Iql = (−1)
wp(p−2q)
R
p−2q
Z
Iqkj = (−1)
hpi
2
(l)
(x) f (x) dx, if 0 ≤ l ≤ q ≤
,
2
wp(p−2q) (x) Re rqkj f (k) (x) f (j) (x) dx,
R
if 0 ≤ k < j ≤ q ≤
where
rqkj = (−1)q−
k+j
2
∈ {±1, ±i}
hpi
2
,
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
and
p
wp = (x − xm ) w.
We assume that all these integrals exist. Finally we denote
Dq =
q
X
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X
Aql Iql + 2
l=0
if |Dq | < ∞ holds for 0 ≤ q ≤
Bqkj Iqkj ,
0≤k<j≤q
p
, where
2
2
q
Aql =
β 2(q−l) ,
l
q
q
Bqkj = sqk
β 2q−j−k ,
k
j
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with β = 2πξm , and sqk = (−1)q−k , and
[p/2]
Ep,f =
X
Cq Dq ,
q=0
if |Ep,f | < ∞ holds for p ∈ N.
Besides we assume the two conditions:
p−2q−1
X
(8.1)
2 (p−2q−r−1)
(−1)r lim wp(r) (x) f (l) (x)
= 0,
|x|→∞
r=0
for 0 ≤ l ≤ q ≤
John Michael Rassias
p
, and
2
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p−2q−1
(8.2)
(p−2q−r−1)
X
r
(r)
(k)
(j)
(−1) lim wp (x) Re rqkj f (x) f (x)
= 0,
r=0
|x|→∞
for 0 ≤ k < j ≤ q ≤ p2 . From these preliminaries we establish the following
Heisenberg-Pauli-Weyl inequality .
2
Theorem 8.1. If f ∈ L (R) , then
(8.3)
On the Heisenberg-Pauli-Weyl
Inequality
q
2p
(µ2p )w,|f |2
q
2p
1
√
(µ2p ) fˆ 2 ≥
| |
2π p 2
holds for any fixed but arbitrary p ∈ N.
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p
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Equality holds in (8.3) iff the a-differential equation
fa(p) (x) = −2cp (x − xm )p fa (x)
of pth order holds for constants cp > 0, and any fixed but arbitrary p ∈ N.
In fact, if cp =
holds iff
f (x) = e2πixξm
kp2
2
> 0, kp ∈ R−{0}, p ∈ N, then this a-differential equation
p−1
X
j=0
"
aj xjδ 1 +
∞
X
(−1)m+1
m=0
kp2 x2p
δ
m+1
2(m+1)p+j
Pp2p+j Pp4p+j · · · Pp
#
,
√
holds, where xδ = x − xm 6= 0, i = −1, xm , ξm (∈ R) are any fixed but
arbitrary constants, and aj (j = 0, 1, 2, . . . , p − 1) are arbitrary constants in C,
as well as
Pp2(m+1)p+j = (2 (m + 1) p + j) (2 (m + 1) p + j − 1) · · · ((2m + 1) p + j + 1) ,
denote permutations for p ∈ N, m ∈ N0 , and j = 0, 1, 2, . . . , p − 1.
We note that if xm is the mean of x for |f |2 then
Z ∞
Z
2
2
2
xm =
x |f (x)| dx =
x |f (x)| − |f (−x)| dx .
2
Thus if f is either odd
2 or even, then |f | is even and xm = 0. Similarly, if ξm is
the mean of ξ for fˆ , then
Z
ξm =
R
John Michael Rassias
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R
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Inequality
ˆ 2
ξ f (ξ) dξ.
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Also ξm = 0 if fˆ is either odd or even.
We also note that the conditions (8.1) – (8.2) may be replaced by the two
conditions:
2 (p−2q−r−1)
(8.4)
lim wp(r) (x) f (l) (x)
= 0,
|x|→∞
for 0 ≤ l ≤ q ≤
(8.5)
p
2
and 0 ≤ r ≤ p − 2q − 1, and
(p−2q−r−1)
lim wp(r) (x) Re rqkj f (k) (x) f (j) (x)
= 0,
On the Heisenberg-Pauli-Weyl
Inequality
|x|→∞
John Michael Rassias
for 0 ≤ k < j ≤ q ≤
p
2
and 0 ≤ r ≤ p − 2q − 1.
Proof of the Theorem. In fact, from the generalized Plancherel-Parseval-Rayleigh
identity (4.1), and the fact that |eax | = 1 as a = −2πξm i, one gets
(8.6) Mp = (µ2p )w,|f |2 · (µ2p ) fˆ 2
| |
Z
Z
2 2p
2
2p ˆ
2
=
w (x) (x − xm ) |f (x)| dx
(ξ − ξm ) f (ξ) dξ
R
R
Z
Z
(p)
2
1
2p
2
2
fa (x) dx .
w (x) (x − xm ) |fa (x)| dx
=
(2π)2p
R
R
From (8.6) and the Cauchy-Schwarz inequality, we find
Z
2
1
(p)
wp (x) fa (x) fa (x) dx ,
(8.7)
Mp ≥
(2π)2p
R
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where wp = (x − xm )p w, and fa = eax f .
From (8.7) and the complex inequality
|ab| ≥
(8.8)
1
ab + ab
2
(p)
with a = wp (x) fa (x), b = fa (x), we get
Z
2
1
1
(p)
(p)
(8.9) Mp ≥
wp (x) fa (x) fa (x) + fa (x)fa (x) dx .
(2π)2p 2 R
On the Heisenberg-Pauli-Weyl
Inequality
From (8.9) and the generalized differential identity (*), one finds


 2
Z
[p/2]
p−2q X
d
1
2
(8.10)
Mp ≥ 2(p+1) 2p  wp (x) 
Cq p−2q fa(q) (x)  dx .
2
π
dx
R
q=0
John Michael Rassias
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JJ
J
From (8.10) and the Lagrange type differential identity (7.1), we find
(8.11) Mp ≥
1
22(p+1) π 2p


Z
[p/2]
X
dp−2q
 wp (x) 
Cq p−2q
dx
R
q=0
+2
X
q
X
2
Aql f (l) (x)
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l=0
Bqkj Re rqkj f (k) (x) f (j) (x)
II
I
Close
#2
!#
dx
.
0≤k<j≤q
From the generalized integral identity (6.2), from f ∈ L2 (R), the two conditions
(8.1) – (8.2) (or (8.4) – (8.5)), or from (6.4) – (6.5), and that all the integrals
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J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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exist, one gets
Z
wp (x)
(8.12)
R
2
dp−2q (l)
dx
f
(x)
dxp−2q
Z
2
p−2q
= (−1)
wp(p−2q) (x) f (l) (x) dx
R
= Iql ,
as well as
Z
dp−2q
(8.13)
wp (x) p−2q Re rqkj f (k) (x) f (j) (x)
dx
R
Z
p−2q
= (−1)
wp(p−2q) (x) Re rqkj f (k) (x) f (j) (x) = Iqkj .
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Title Page
R
From (8.11) and (8.12) – (8.13) we find the generalized 2pth order moment
Heisenberg uncertainty inequality (for p ∈ N)

!2
[p/2]
q
X
X
X
1
(Hp )
Mp ≥ 2(p+1) 2p 
Cq
Aql Iql + 2
Bqkj Iqkj 
2
π
q=0
l=0
0≤k<j≤q
=
1
E 2 , (Hp ),
22(p+1) π 2p p,f
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where
[p/2]
Ep,f =
X
q=0
Cq Dq , if |Ep,f | < ∞
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holds, or the general moment uncertainty formula
q
p
1 p
2p
√
(8.14)
Mp ≥
|Ep,f |.
2π p 2
(p)
Equality holds in (8.3) iff the a-differential equation fa (x) = −2cp xpδ fa (x) of
pth order with respect to x holds for some constant cp = 12 kp2 > 0, kp ∈ R−{0},
and any fixed but arbitrary p ∈ N.
We consider the general a-differential equation
(ap )
dp y
+ kp2 xpδ y = 0
p
dx
of pth order, where xδ = x − xm 6= 0, kp 6= 0, y = fa (x) = eax f (x),
a = −2πξm i, p ∈ N, and the equivalent δ-differential equation
(δp )
because
and
dp y
+ kp2 xpδ y = 0,
dxpδ
dy
dy dxδ
dy d (x − xm )
dy
=
=
=
,
dx
dxδ dx
dxδ
dx
dxδ
dp y
dp y
=
,
dxp
dxpδ
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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p ∈ N.
In order to solve equation (δp ) one may employ the following power series
method ([18], [21], [25]) in (δp ). In fact, we consider the power series expansion
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y=
P∞
n=0
an xnδ about xδ0 = 0, converging (absolutely) in
an = ∞, xδ 6= 0.
|xδ | < ρ = lim n→∞ an+1 Thus
kp2 xpδ y
=
∞
X
kp2 an xn+p
,
δ
n=0
and
p
dy
=
dxpδ
∞
X
Ppn an xn−p
δ
n=p
(with permutations Ppn = n (n − 1) (n − 2) · · · (n − p + 1))
∞
X
=
Ppn+2p an+2p xn+p
δ
n+2p=p
(or n=−p)
(with n + 2p) on n above and
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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Ppn+2p = (n + 2p) (n + 2p − 1) (n + 2p − 2) · · · (n + p + 1))
=
=
−1
X
n=−p
∞
X
Ppn+2p an+2p xδn+p +
Ppn+2p an+2p xn+p
δ
n=0
∞
X
n=0
Ppn+2p an+2p xn+p
δ
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(with an+2p = 0 for all n ∈ { − p, −(p − 1), . . . , −1}, or equivalently ap =
ap+1 = · · · = a2p−1 = 0).
Therefore from these and equation (δp ) one gets the following recursive relation
Ppn+2p an+2p + kp2 an = 0,
or
an+2p = −
(Rp )
kp2 an
Ppn+2p
,
n ∈ N0 , p ∈ N.
From the “null condition”
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
ap = 0, ap−1 = 0, . . . , a2p−1 = 0
Title Page
and (Rp ) we get
a3p = a5p = a7p = · · · = 0, a3p+1 = a5p+1 = a7p+1 = · · · = 0, . . . ,
and
a4p−1 = a6p−1 = a8p−1 = · · · = 0,
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respectively. From (Rp ) and n = 2pm with m ∈ N0 one finds the following p
sequences
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(a2pm+2p ) , (a2pm+2p+1 ) , (a2pm+2p+2 ) , . . . , (a2pm+3p−1 ) ,
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with fixed p ∈ N and all m ∈ N0 , such that
a2p = −
kp2
2p a0 ,
Pp
a4p = −
kp2
4p a2p =
Pp
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kp4
a0 , . . . ,
Pp2p Pp4p
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a2pm+2p = (−1)m+1
kp2m+2
Pp2p Pp4p · · · Pp2pm+2p
a0
are the elements of the first sequence (a2pm+2p ) in terms of a0 , and
a2p+1 = −
kp2
2p+1 a1 ,
Pp
a4p+1 = −
a2pm+2p+1 = (−1)m+1
kp2
4p+1 a2p+1 =
Pp
kp4
Pp2p+1 Pp4p+1
kp2m+2
Pp2p+1 Pp4p+1 · · · Pp2pm+2p+1
a1 , . . . ,
a1
On the Heisenberg-Pauli-Weyl
Inequality
are the elements of the second sequence (a2pm+2p+1 ) in terms of a1 , . . ., and
kp2
a2p+(p−1) = a3p−1 = − 3p−1 ap−1 ,
Pp
John Michael Rassias
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a2p+(3p−1) = a5p−1 =
a2pm+(3p−1)
kp2
− 5p−1 a3p−1
Pp
= a(2m+3)p−1 = (−1)m+1
=
kp2
ap−1 , . . . ,
Pp3p−1 Pp5p−1
kp2m+2
a
(2m+3)p−1 p−1
Pp3p−1 Pp5p−1 · · · Pp
are the elements of the last sequence (a2pm+3p−1 ) in terms of ap−1 . Therefore
we find the following p solutions
y0 = y0 (xδ ) = 1 +
∞
X
m=0
(−1)m+1
kp2 x2p
δ
2(m+1)p
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m+1
Pp2p Pp4p · · · Pp
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,
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"
y1 = y1 (xδ ) = xδ 1 +
∞
X
(−1)m+1
m=0
"
yp−1 = yp−1 (xδ ) =
xp−1
δ
1+
∞
X
kp2 x2p
δ
m+1
#
2(m+1)p+1
Pp2p+1 Pp4p+1 · · · Pp
kp2 x2p
δ
m+1
(−1)
m=0
,...,
m+1
#
(2m+3)p−1
Pp3p−1 Pp5p−1 · · · Pp
,
or equivalently the
"
yj = yj (xδ ) =
xjδ
1+
∞
X
m+1
(−1)
m=0
kp2 x2p
δ
m+1
#
2(m+1)p+j
Pp2p+j Pp4p+j · · · Pp
for all j ∈ {0, 1, . . . , p − 1}, of the differential equation (δp ), in the form of
power series converging (absolutely) by the ratio test.
Thus an arbitrary solution of (δp ) (and of (ap )) is of the form
" p−1
#
X
f (x) = e−ax
aj yj (xδ ) ,
xδ 6= 0,
j=0
with arbitrary constants aj (j = 0, 1, 2, . . . , p − 1).
Choosing
a0 = 1, a1 = 0, a2 = 0, . . . , ap−2 = 0, ap−1 = 0;
a0 = 0, a1 = 1, a2 = 0, . . . , ap−2 = 0, ap−1 = 0;
··· ;
and
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
a0 = 0, a1 = 0, a2 = 0, . . . , ap−2 = 0, ap−1 = 1,
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one gets that yi (i ∈ {0, 1, 2, . . . , p − 1}) are partial solutions of (δp ), satisfying the initial conditions
y0 (0) = 1,
(p−1)
y00 (0) = 0, . . . , y0
(0) = 0;
(p−1)
y1 (0) = 0, y10 (0) = 1, . . . , y1
(0) = 0;
······························ ;
On the Heisenberg-Pauli-Weyl
Inequality
and
(p−1)
0
yp−1 (0) = 0, yp−1
(0) = 0, . . . , yp−1 (0) = 1.
If Yp = (y0 , y1 , . . . , yp−1 ), then the Wronskian at xδ
y0 (0)
y1 (0)
···
0
0
y (0)
y1 (0)
···
W (Yp (0)) = 0
·
···
···
· ·(p−1)
(p−1)
y
(0) y1
(0) · · ·
0
1 0 ··· 0 0 1 ··· 0 = ..
.. = 1 6= 0,
.
.
.
. . 0 0 ··· 1 John Michael Rassias
= 0 is
yp−1 (0)
y 0 p−1 (0)
···
(p−1)
yp−1 (0)
yielding that these p partial solutions y0 , y1 , . . . , yP
p−1 of (δp ) are linearly independent. Thus the above formula y = fa (x) = p−1
j=0 aj yj gives the general
solution of the equation (δp ) (and also of (ap )).
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We note that both the above-mentioned differential equations (ap ) and (δp )
are solved completely via well-known special functions for p = 1 (with Gaussian functions) and for p = 2 (with Bessel functions), and via functions in terms
of power series converging in R for p ≥ 3. Therefore the proof of our theorem
is complete.
We claim that, if p = 1, the function f : R → C given explicitly in our
Introduction (with c = c1 = k12 /2 > 0, k1 ∈ R − {0}) satisfies the equality of
(H1 ). In fact, the corresponding a-differential equation
(a1 )
where xδ = x − xm
dy
+ k12 xδ y = 0,
dx
6= 0, y = fa (x) = eax f (x), a = −2πξm i, is satisfied
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
by
Title Page
y0 = y0 (x)
∞
X
=1+
(−1)m+1
=1+
m=0
∞
X
m+1
(k12 x2δ )
2(m+1)
P12 P14 · · · P1
m+1
(−1)m+1
m=0
(c1 x2δ )
(m + 1)!
2(m+1)
or
∞
X
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(by the power series method and because
P12 P14 · · · P1
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= 2 · 4 · · · · 2 (m + 1) = 2m+1 (m + 1)!),
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m
(cx2δ )
y0 = 1 +
(−1)
m!
m=1
m
−cx2δ
=1+ e
2
− 1 = e−cxδ
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P
m tm
−t
(because ∞
− 1).
m=1 (−1) m! = e
Therefore the general solution of the differential equation (a1 ) is of the form
y = a0 y0 (with arbitrary constant c0 = a0 ), or
2
2
eax f (x) = c0 e−c(x−xm ) , or f (x) = c0 e2πixξm · e−c(x−xm ) .
However, one may establish this f much faster, by the direct application of
the method of separation of variables to the differential equation (a1 ).
Analogously to the proof of (H1 ) in the Introduction we prove the following
more general inequality (H2 ).
8.1.
Fourth Order Moment Heisenberg Inequality
For any f ∈ L2 (R), f : R → C and any fixed but arbitrary constants xm ,
ξm ∈ R, the fourth order moment Heisenberg inequality
(µ4 )|f |2
(H2 )
1
· (µ4 ) fˆ 2 ≥
E2 ,
| |
64π 4 2,f
holds, if
Z
x4δ |f (x)|2 dx
(µ4 )|f |2 =
R
and
Z
(µ4 ) fˆ 2 =
| |
R
John Michael Rassias
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2
4ˆ
ξδ f (ξ) dξ
with xδ = x − xm , and ξδ = ξ − ξm , are the fourth order moments, and
Z
Z
√
−2iπξx
ˆ
f (ξ) =
e
f (x)dx, f (x) =
e2iπξx fˆ (ξ)dξ,
i = −1,
R
On the Heisenberg-Pauli-Weyl
Inequality
R
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as well as
E2,f = 2
Z h
2
2 2
1 − 4π 2 ξm
xδ |f (x)|2 − x2δ |f 0 (x)|
R
i
−4πξm x2δ Im f (x) f 0 (x) dx,
if |E2,f | < ∞ holds, where Im (·) denotes the imaginary part of (·).
Equality holds in (H2 ) iff the a-differential equation fa00 (x) = −2c2 x2δ fa (x) of
second order holds, for a = −2πξm i, y = fa (x) = eax f (x) and a constant
c2 = 12 k22 > 0, k2 ∈ R − {0}, xδ = x − xm 6= 0, or equivalently
d2 y
+ k22 x2δ y = 0
dx2
(a2 )
holds iff
John Michael Rassias
Title Page
p
1
1
2πixξm
2
2
f (x) = |xδ |e
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
2
2
holds for some constants c20 , c21 ∈ C and with J±1/4 the Bessel functions of the
first kind of orders ± 14 , [16]. We note that if xm 6= 0 and ξm = 0, then
p
1
1
2
2
f (x) = |xδ | c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
.
2
2
We claim that the above function f in terms of the Bessel functions J±1/4 is the
general solution of the said a-differential equation (a2 ) of second order. In fact,
the δ-differential equation
(δ2 )
On the Heisenberg-Pauli-Weyl
Inequality
d2 y
+ k22 x2δ y = 0
2
dxδ
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is equivalent to the following Bessel equation
"
2 #
2
d
u
1
du
z2 2 + z
+ z2 −
u=0
dz
dz
4
of order r = 41 , with u = √y
and z =
|xδ |
1
2
|k2 | x2δ . But the general solution of
this Bessel equation is
.p
u (z) = y
|xδ | = c20 J−1/4 (z) + c21 J1/4 (z)
1
1
2
2
= c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
2
2
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
for some constants c20 , c21 ∈ C.
In fact, if we denote
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(
S = sgn (xδ ) =
(8.15)
Contents
1,
xδ > 0
JJ
J
,
−1, xδ < 0
then one gets
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du
du
=
dz
dxδ
dz
=
dxδ
"
S
3/2
|xδ |
dy
1 y
−
dxδ 2 |xδ |5/2
#,
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|k2 | ,
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and
d2 u
d
=
2
dz
dxδ
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Page 60 of 151
du
dz
dz
=
dxδ
"
d2 y
S dy
5 y
−2
+
5/2 dx2
7/2 dx
4 |xδ |9/2
δ
|xδ |
|xδ |
δ
S
#,
k22 .
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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Thus we establish
z2
2
d2 u
du
1
1
3/2 d y
+
z
=
|x
|
+ u.
δ
2
2
dz
dz
4
dxδ 16
But
d2 y
= −k22 x2δ y,
2
dxδ
or
|xδ |3/2
d2 y
y
= −k22 x4δ p
= −4z 2 u.
2
dxδ
|xδ |
Therefore the above-mentioned Bessel equation holds.
However,
dy
dy dxδ
dy d (x − xm )
dy
=
=
=
,
dx
dxδ dx
dxδ
dx
dxδ
and
d2 y
d dy
d
dy dxδ
d2 y
=
=
=
.
dx2
dx dx
dxδ dxδ dx
dx2δ
Therefore the above two equations (a2 ) and (δ2 ) are equivalent. Thus one gets
p
1
1
ax
2
2
y = fa (x) = e f (x) = |xδ | c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
2
2
or
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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p
1
1
2
2
−ax
f (x) = |xδ |e
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
2
2
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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establishing the function f in terms of the two Bessel functions J±1/4 .
However,
∞
z 2n
z 14 X
n
2
, z > 0.
J1/4 (z) =
(−1)
2 n=0
n!Γ 14 + n + 1
Thus, if z = 12 |k2 | x2δ > 0, then
√
p
2
1
2p
k22 5
4
2
|xδ |J1/4
|k2 | xδ = S 1 |k2 | xδ −
x + ··· ,
2
4·5 δ
Γ 4
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
because
1
1
9
5
1
5
Γ
= Γ
,Γ
= Γ
,...,
4
4
4
4
16
4
and S = sgn (xδ ), xδ 6= 0, such that |xδ | = Sxδ . Similarly,
∞
z 2n
z − 14 X
n
2
, z > 0.
J−1/4 (z) =
(−1)
1
2
n!Γ
−
+n+1
4
n=0
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Therefore
p
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|xδ |J−1/4
1
|k2 | x2δ
2
=
Γ
1
4
π
1
k22 4
p
1−
x + ··· ,
4
3·4 δ
|k2 |
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because
√
3
1
1
π
1
Γ
Γ
=Γ
Γ 1−
=
= π 2,
1
4
4
4
4
sin 4 π
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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or
√
√
3
π 2
7
3
3
3
3 2π
, and Γ
, . . ..
Γ
=
=Γ 1+
= Γ
=
4
4
4
4
4
Γ 14
4Γ 14
A direct way to find the general solution of the above δ-differential equation
(δ2 ) is by applying the power series method for (δ2 ). In fact, consider two
∞
P
arbitrary constants a0 = c20 and a1 = c21 such that y =
an xnδ , about xδ0 =
n=0
0, converging (absolutely) in
an = ∞,
|xδ | < ρ = lim n→∞ an+1 xδ 6= 0.
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Title Page
Thus
dy
=
dxδ
∞
X
n=1
nan xn−1
=
δ
∞
X
(n + 1) an+1 xnδ ,
n=0
and
∞
d2 y X
=
(n + 1) nan+1 xn−1
δ
dx2δ
n=1
=
=
∞
X
(n + 2) (n + 1) an+2 xnδ
n=0
∞
X
n+2=0 or
n=−2
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(n + 4) (n + 3) an+4 xn+2
δ
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= (2a2 + 6a3 xδ ) +
∞
X
,
(n + 4) (n + 3) an+4 xn+2
δ
n=0
as well as
k22 x2δ y
=
kδ2
∞
X
an xn+2
.
δ
n=0
Therefore from (δ2 ) one gets the recursive relation
an+4 = −
(R2 )
k22 an
,
(n + 4) (n + 3)
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
with a2 = a3 = 0.
Letting n = 4m with m ∈ N0 = {0} ∪ N in this recursive relation, we find
the following two solutions of the equation (δ2 ):
y0 = y0 (xδ )
∞
X
=1+
(−1)m+1
=1−
m=0
k22
3·4
x4δ +
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m+1
(k22 x4δ )
3 · 4 · · · (4m + 3) (4m + 4)
k24
k26
x8δ −
x12
+ ··· ,
3·4·7·8
3 · 4 · 7 · 8 · 11 · 12 δ
and
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y1 = y1 (xδ )
"
= xδ 1 +
Page 64 of 151
∞
X
m=0
m+1
m+1
(−1)
(k22 x4δ )
4 · 5 · · · (4m + 4) (4m + 5)
#
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= xδ −
k22 5
k24
k26
xδ +
x9δ −
x13 + · · · .
4·5
4·5·8·9
4 · 5 · 8 · 9 · 12 · 13 δ
We note that each one of these two power series converges by the ratio test.
Thus an arbitrary solution of (δ2 ) (and of (a2 )) is of the form
y = c20 y0 + c21 y1 ,
or
f (x) = e−ax [c20 y0 (xδ ) + c21 y1 (xδ )] ,
where xδ = x − xm 6= 0,
h p p
π 4 |k2 | |xδ |J−1/4
y0 =
Γ 14
1
2
|k2 | x2δ
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
i
,
S ∈ { ± 1},
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and
h
y1 =
SΓ
1
4
p
1
2
|xδ |J1/4
√ p
2 2 4 |k2 |
|k2 | x2δ
i
,
where S is defined by (8.15).
Besides we note that from a2 = 0, a3 = 0 and the above recursive relation (R2 )
one gets a6 = a10 = a14 = · · · = 0, as well as a7 = a11 = a15 = · · · = 0,
respectively.
From this recursive relation (R2 ) and n = 4m with m ∈ N0 we get the
following two sequences (a4m+4 ), (a4m+5 ), such that
a4 = −
k22
k2
k24
a0 , a8 = − 2 a4 =
a0 , . . . ,
3·4
7·8
3·4·7·8
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a4m+4 = (−1)m+1
k22m+2
a0 ,
3 · 4 · · · · · (4m + 3) (4m + 4)
and
k22
k2
k24
a1 , a9 = − 2 a5 =
a1 , . . . ,
4·5
8·9
4·5·8·9
k22m+2
a4m+5 = (−1)m+1
a1 .
4 · 5 · · · · · (4m + 4) (4m + 5)
Choosing a0 = c20 = 1, a1 = c21 = 0; and a0 = c20 = 0, a1 = c21 = 1, one
gets that y0 and y1 are partial solutions of (δ2 ), satisfying the initial conditions
a5 = −
y0 (0) = 1, y00 (0) = 0; and y1 (0) = 0, y10 (0) = 1.
Therefore the Wronskian of y0 , y1 at xδ = 0 is
1
y0 (0)
y
(0)
1
=
W (y0 , y1 ) (0) = y00 (0)
y10 (0) 0
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Title Page
0 = 1 6= 0,
1 yielding that these p = 2 solutions y0 , y1 are linearly independent. We note that,
if we divide the above power series (expansion) solutions y0 and y1 , we have
y1 (xδ )
= y1 (xδ ) (y0 (xδ ))−1
y0 (xδ )
−1
k22 4
k22 5
1 − xδ + · · ·
= xδ − xδ + · · ·
20
12
2
k
= xδ + 2 x5δ + · · · ,
30
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which obviously is nonconstant, implying also that y0 and y1 are linearly independent.
Thus the above formula y = a0 y0 + a1 y1 gives the general solution of (δ2 )
(and also of (a2 )). Similarly we prove the following inequality (H3 ).
8.2.
Sixth Order Moment Heisenberg Inequality
For any f ∈ L2 (R), f : R → C and any fixed but arbitrary constants xm ,
ξm ∈ R, the sixth order moment Heisenberg inequality
1
(H3 )
(µ6 )|f |2 · (µ6 ) fˆ 2 ≥
E2 ,
| |
256π 6 3,f
holds, if
Z
x6δ |f (x)|2 dx
(µ6 )|f |2 =
R
and
Z
(µ6 ) fˆ 2 =
| |
R
R
as well as
E3,f = −3
John Michael Rassias
Title Page
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2
ξδ6 fˆ (ξ) dξ
with xδ = x − xm , and ξδ = ξ − ξm , and
Z
Z
−2iπξx
ˆ
f (ξ) =
e
f (x)dx, f (x) =
e2iπξx fˆ (ξ)dξ,
R
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Z h
R
2
2 2
2 1 − 6π 2 ξm
xδ |f (x)|2 − 3x2δ |f 0 (x)|
i
−12πξm x2δ Im f (x) f 0 (x) dx,
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if |E3,f | < ∞ holds, where Im (·) denotes the imaginary part of (·).
Equality holds in (H3 ) iff the a-differential√equation fa000 (x) = −2c3 x3δ fa (x)
of third order holds, for a = −2πξm i, i = −1, fa = eax f , and a constant
k2
c3 = 23 > 0, k3 ∈ R, or equivalently iff
f (x) = e2πixξm
2
X
"
aj xjδ 1 +
∞
X
(−1)m+1
m=0
j=0
m+1
(k32 x6δ )
×
(4 + j) (5 + j) · · · (6m + 5 + j) (6m + 6 + j)
#
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
holds, where xδ 6= 0, and aj (j = 0, 1, 2) are arbitrary constants in C.
Consider the a-differential equation
3
(a3 )
dy
+ k32 x3δ y = 0,
3
dx
with y = fa (x) and the equivalent δ-differential equation
(δ3 )
d3 y
+ k32 x3δ y = 0,
dx3δ
with xδ = x − xm and k3 ∈ R − {0}, such that d3 y/ dx3 = d3 y/ dx3δ .
Employing the
method for (δ3 ), one considers the power series
P∞power series
n
expansion y = n=0 an xδ about xδ0 = 0, converging (absolutely) in
an = ∞,
|xδ | < ρ = lim xδ 6= 0.
n→∞ an+1 Title Page
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Thus
k32 x3δ y
=
∞
X
,
k32 an xn+3
δ
n=0
and
∞
d3 y X
=
n (n − 1) (n − 2) an xn−3
δ
dx3δ
n=3
=
∞
X
(n + 6) (n + 5) (n + 4) an+6 xn+3
δ
On the Heisenberg-Pauli-Weyl
Inequality
n=−3
= 6a3 + 24a4 xδ +
60a5 x2δ
+
∞
X
John Michael Rassias
P3n+6 an+6 xn+3
δ
n=0
=
∞
X
Title Page
P3n+6 an+6 xn+3
δ
Contents
n=0
P3n+6
(with a3 = a4 = a5 = 0), where
= (n + 6) (n + 5) (n + 4). Therefore
from these and equation (δ3 ) one gets the recursive relation
(R3 )
an+6
k 2 an
= − 3n+6 ,
P3
n ∈ N0
a3 = 0,
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From “the null condition”
(N3 )
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a4 = 0,
and
a5 = 0
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and the above recursive relation (R3 ) we get
a9 = a15 = a21 = · · · = 0,
a10 = a16 = a22 = · · · = 0,
a11 = a17 = a23 = · · · = 0
and
respectively.
From (R3 ) and n = 6m with m ∈ N0 one finds the following three sequences
(a6m+6 ), (a6m+7 ), (a6m+8 ), such that
k32
a0 ,
4·5·6
k32
k34
a12 = −
a6 =
a0 ,
10 · 11 · 12
4 · 5 · 6 · 10 · 11 · 12
...,
a6 = −
m+1
a6m+6 = (−1)
k32m+2
a0 ,
4 · 5 · 6 · · · · · (6m + 4) (6m + 5) (6m + 6)
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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and
k32
a1 ,
5·6·7
k32
k34
a13 = −
a7 =
a1 ,
11 · 12 · 13
5 · 6 · 7 · 11 · 12 · 13
...,
a7 = −
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a6m+7 = (−1)m+1
k32m+2
a1 ,
5 · 6 · 7 · · · · · (6m + 5) (6m + 6) (6m + 7)
as well as
k32
a2 ,
6·7·8
k34
k32
a14 = −
a8 =
a2 ,
12 · 13 · 14
6 · 7 · 8 · 12 · 13 · 14
...,
a8 = −
On the Heisenberg-Pauli-Weyl
Inequality
k32m+2
a2 .
6 · 7 · 8 · · · · · (6m + 6) (6m + 7) (6m + 8)
Therefore we find the following three solutions
a6m+8 = (−1)m+1
y0 = y0 (xδ )
∞
X
=1+
(−1)m+1
=1+
m=0
∞ X
m=0
John Michael Rassias
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m+1
(k32 x6δ )
JJ
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4 · 5 · 6 · · · · · (6m + 4) (6m + 5) (6m + 6)
a6m+6
x6m+6
δ
a0
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for a0 6= 0,
y1 = y1 (xδ )
"
= xδ 1 +
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∞
X
m=0
m+1
m+1
(−1)
(k32 x6δ )
5 · 6 · 7 · · · · · (6m + 5) (6m + 6) (6m + 7)
#
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= xδ +
∞ X
a6m+7
m=0
a1
x6m+7
δ
for a1 6= 0, and
y2 = y2 (xδ )
"
= x2δ 1 +
∞
X
m+1
(−1)m+1
m=0
= x2δ +
∞ X
a6m+8
m=0
a2
(k32 x6δ )
6 · 7 · 8 · · · · · (6m + 6) (6m + 7) (6m + 8)
#
x6m+8
δ
John Michael Rassias
for a2 6= 0, of the differential equation (δ3 ), in the form of power series converging (absolutely) by the ratio test. Thus an arbitrary solution of (δ3 ) (and of
(a3 )) is of the form
y = eax f (x) = a0 y0 + a1 y1 + a2 y2 ,
or
f (x) = e−ax [a0 y0 (xδ ) + a1 y1 (xδ ) + a2 y2 (xδ )] ,
xδ 6= 0, with arbitrary constants ai (i = 0, 1, 2) in C. Choosing
a0 = 1,
a0 = 0,
a0 = 0,
a1 = 0,
a1 = 1,
a1 = 0,
On the Heisenberg-Pauli-Weyl
Inequality
a2 = 0;
a2 = 0; and
a2 = 1,
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one gets that yj (j = 0, 1, 2) are partial solutions of (δ3 ), satisfying the initial
conditions
y0 (0) = 1,
y1 (0) = 0,
y2 (0) = 0,
y00 (0) = 0, y000 (0) = 0;
y10 (0) = 1, y100 (0) = 0;
y20 (0) = 0, y200 (0) = 1.
and
Therefore the Wronskian of yj (j = 0, 1, 2) at xδ = 0 is
y0 (0)
y1 (0)
y2 (0)
y10 (0)
y20 (0)
W (y0 , y1 , y2 ) (0) = y00 (0)
00
y (0)
y100 (0)
y200 (0)
0
1
0
0 1
0 = 1 6= 0,
= 0
0
0
1 yielding that these p = 3 partial solutionsP
yj (j = 0, 1, 2) of (δ3 ) are linearly
independent. Thus the above formula y = 2j=0 aj yj gives the general solution
of (δ3 ) (and also of (a3 )). Analogously we establish the following inequality
(H4 ):
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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8.3.
Eighth Order Moment Heisenberg Inequality
For any f ∈ L2 (R), f : R → C and any fixed but arbitrary constants xm ,
ξm ∈ R, the eighth order moment Heisenberg inequality
1
(µ8 )|f |2 · (µ8 ) fˆ 2 ≥
E2 ,
| |
1024π 8 4,f
(H4 )
holds, if
Z
x8δ |f (x)|2 dx
(µ8 )|f |2 =
On the Heisenberg-Pauli-Weyl
Inequality
R
and
Z
(µ8 ) fˆ 2 =
| |
R
ξδ8
John Michael Rassias
ˆ 2
f (ξ) dξ
with xδ = x − xm , ξδ = ξ − ξm , and
Z
Z
−2iπξx
ˆ
f (ξ) =
e
f (x)dx, f (x) =
e2iπξx fˆ (ξ)dξ,
R
R
as well as
Z
E4,f = 2
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4 4
2 2
xδ |f (x)|2
4 3 − 24π 2 ξm
xδ + 16π 4 ξm
Close
R
2
2
i
2
xδ |f 0 (x)| + x4δ |f 00 (x)|
−8x2δ 3 − 2π 2 ξm
h
2 2
−8πξm x2δ 4 3 − π 2 ξm
xδ Im f (x) f 0 (x)
io
dx,
+πξm x2δ Re f (x) f 00 (x) − x2δ Im f 0 (x) f 00 (x)
2
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if |E4,f | < ∞ holds, where Re (·) and Im (··) denote the real part of (·) and the
imaginary part of (··), respectively.
Equality holds in (H4 ) iff the a-differential equation
fa(4) (x) = −2c4 x4δ fa (x)
√
of fourth order holds, for a = −2πξm i, i = −1, fa = eax f , and a constant
k2
c4 = 24 > 0, k4 ∈ R, or equivalently iff
f (x) = e2πixξm
3
X
j=0
"
aj xjδ 1 +
∞
X
On the Heisenberg-Pauli-Weyl
Inequality
(−1)m+1
John Michael Rassias
m=0
m+1
(k42 x8δ )
×
(5 + j) (6 + j) · · · (8m + 7 + j) (8m + 8 + j)
holds, where xδ 6= 0, and aj (j = 0, 1, 2, 3) are arbitrary constants in C.
#
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8.4.
First Four Generalized Weighted Moment Inequalities
(i)
(8.16) M1 = (µ2 )w,|f |2 (µ2 ) fˆ 2
| |
Z
Z
2 2
2
2ˆ
2
w (x) (x − xm ) |f (x)| dx
(ξ − ξm ) f (ξ) dξ
=
R
R
1
≥
(C0 D0 )2
16π 2
1 2
1
=
[C0 (A00 I00 )]2 =
I
2
16π
16π 2 00
Z
2
1
1
(1)
2
=
w1 (x) |f (x)| dx =
E2 ,
2
2 1,f
16π
16π
R
because
I00 = −
Z (1)
w1 |f |2 (x) dx
R
with w1 (x) = w(x)(x − xm ). We note that if w = 1 then
E1,f = C0 D0 = I00
Z (1)
=−
w1 |f |2 (x) dx
ZR
Z 2
2
= − |f (x)| dx = −E|f |2 = −1 = −E fˆ 2 = − fˆ (ξ) dξ
| |
R
R
by the Plancherel-Parseval-Rayleigh identity, if |E1,f | < ∞ holds. Thus from
(8.16) one gets the classical second order moment Heisenberg uncertainty principle which says that the product of the variance (µ2 )|f |2 of x for the probability
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Inequality
John Michael Rassias
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2
density |f |2 and the variance (µ2 ) fˆ 2 of ξ for the probability density fˆ is at
| |
E2
2
|f |
least 16π
2 , which is the second order moment Heisenberg Inequality (H1 ) in
1
our Introduction. The Heisenberg lower bound H ∗ = 4π
, for E|f |2 = 1, can
be different if one chooses a different formula for the Fourier transform fˆ of f .
Finally, the above inequality (8.16) generalizes (H1 ) of our Introduction (there
w = 1).
(ii)
(8.17)
M2 = (µ4 )w,|f |2 (µ4 ) fˆ 2
| |
Z
Z
2 4
2
4ˆ
2
=
w (x) (x − xm ) |f (x)| dx
(ξ − ξm ) f (ξ) dξ
R
R
1
(C0 D0 + C1 D1 )2
≥
64π 4
1
=
[C0 (A00 I00 ) + C1 (A10 I10 + A11 I11 + 2B101 I101 )]2
64π 4
2
1 =
I00 − 2 β 2 I10 + I11 − 2βI101
4
64π Z
1
(2)
=
w2 (x) |f (x)|2 dx
4
64π
R
2
Z
h
i
2
2
0 2
0
−2 w2 (x) β |f | + |f | + 2β Im f f¯ (x) dx ,
R
because Re if f¯0 (x) = − Im f f¯0 (x), and
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Inequality
John Michael Rassias
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Z
I00 =
ZR
I10 =
ZR
I11 =
(2)
w2 (x) |f (x)|2 dx,
w2 (x) |f (x)|2 dx,
2
w2 (x) |f 0 (x)| dx,
R
and
Z
I101 =
w2 (x) Re if f¯0 (x) dx,
On the Heisenberg-Pauli-Weyl
Inequality
R
with w2 (x) = w(x)(x − xm )2 .
It is clear that (8.17) is equivalent to
Z 1
(2)
2
2
(8.18) M2 ≥
w
−
2β
w
2 (x) |f (x)| dx
2
64π 4 R
2
Z
Z
2
0
0
¯
−2 w2 (x) |f (x)| dx − 4β w2 (x) Im f f (x) dx
R
=
R
1
E2 ,
64π 4 2,f
p
4
M2 ≥
1
√
2π 2
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where
(8.20)
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(8.19)
John Michael Rassias
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E2,f = C0 D0 + C1 D1
= D0 − 2D1 = I00 − 2 β 2 I10 + I11 − 2βI101
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=
Z h
(2)
w2
2
− 2β w2
0 2
|f | − 2w2 |f | − 4βw2 Im f f¯0
2
i
(x) dx,
R
if
|E2,f | < ∞ holds. We note that if |E2,f | = 21 holds, then from (8.19) one gets
√
4
1
M2 ≥ 4π
(= H ∗ ),while if |E2,f | = 1, then
p
1
1√
4
2 (> H ∗ ).
M2 ≥ √ =
4π
2π 2
√
Thus we observe that the lower bound of 4 M2 is greater than H ∗ if |E2,f | >
1
; the same with H ∗ if |E2,f | = 12 ; and smaller than H ∗ , if 0 < |E2,f | < 12 .
2
Finally, the above inequality (8.18) generalizes (H2 ) of Section 8 (there w = 1).
(iii)
(8.21)
M3 = (µ6 )w,|f |2 (µ6 ) fˆ 2
| |
Z
Z
2 6
2
6ˆ
2
=
w (x) (x − xm ) |f (x)| dx
(ξ − ξm ) f (ξ) dξ
R
R
1
(C0 D0 + C1 D1 )2
256π 6
1
=
[C0 (A00 I00 ) + C1 (A10 I10 + A11 I11 + 2B101 I101 )]2
256π 6
2
1 =
I00 − 3 β 2 I10 + I11 − 2βI101
6
256π Z
1
(3)
=
− w3 (x) |f (x)|2 dx
6
256π
R
2
Z
h
i
(1)
2
2
0 2
0
+ 3 w3 (x) β |f | + |f | + 2β Im f f (x) dx ,
≥
R
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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because
Z
I00 = −
(3)
w3
Z
2
I10 = −
(x) |f (x)| dx,
R
(1)
w3 (x) |f (x)|2 dx,
R
Z
I11 = −
2
(1)
w3 (x) |f 0 (x)| dx,
R
and
Z
(1)
I101 = − w3 (x) Re if (x) f 0 (x) dx
Z R
(1)
=
w3 (x) Im f (x) f 0 (x) dx,
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
R
with w3 (x) = w(x)(x − xm )3 .
It is clear that (8.21) is equivalent to
Z 1
(3)
2 (1)
(8.22) M3 ≥
−w
+
3β
w
(x) |f (x)|2 dx
3
3
256π 6 R
2
Z
Z
2
(1)
(1)
0
0
¯
+3 w3 |f (x)| dx + 6β w3 (x) Im f f (x) dx
R
R
1
=
E2 ,
256π 6 3,f
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or
(8.23)
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1
M3 ≥ √
2π 3 2
p
6
q
3
|E3,f |
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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where
E3,f =
Z h
(3)
(1)
−w3 + 3β 2 w3
(1)
2
(1)
|f |2 + 3w3 |f 0 | + 6βw3 Im f f¯0
i
(x) dx,
R
if
|E3,f | < ∞ holds. We note that if |E3,f | √
= 14 , then from (8.23) we find
√
6
6
1
∗
M3 ≥ 4π (= H ), while if |E3,f | = 1, then M3 ≥ 2π1√
(> H ∗ ). Thus we
3
2
√
observe that the lower bound of 6 M3 is greater than H ∗ if |E3,f | > 14 ; the same
with H ∗ if |E3,f | = 41 ; and smaller than H ∗ , if 0 < |E3,f | < 14 . Finally, the
above inequality (8.22) generalizes (H3 ) of our section 8 (there w = 1).
(iv)
(8.24) M4 = (µ8 )w,|f |2 (µ8 ) fˆ 2
| |
Z
Z
2 8
2
8ˆ
2
=
w (x) (x − xm ) |f (x)| dx
(ξ − ξm ) f (ξ) dξ
R
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Title Page
Contents
R
1
(C0 D0 + C1 D1 + C2 D2 )2
8
1024π
1
=
[C0 (A00 I00 ) + C1 (A10 I10 + A11 I11 + 2B101 I101 )
1024π 8
+ C2 (A20 I20 + A21 I21 + A22 I22
≥
+2B201 I201 + 2B202 I202 + 2B212 I212 )]2
1 =
I00 − 4 β 2 I10 + I11 − 2βI101
8
1024π
2
+2 β 4 I20 + 4β 2 I21 + I22 + 4β 3 I201 + 2β 2 I202 − 4βI212
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Z
1
(4)
=
w4 (x) |f (x)|2 dx
8
1024π
Z R
h
i
(2)
2
2
0 2
0
¯
− 4 w4 (x) β |f | + |f | + 2β Im f f (x) dx
ZR
h
2
2
+ 2 w4 (x) β 4 |f |2 + 4β 2 |f 0 | + |f 00 | + 4β 3 Im f f 0
R
2
2
−2β Re f f 00 + 4β Im f 0 f 00 (x) dx
1
2
,
=
E4,f
8
1024π
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
because
Z
I00 =
ZR
I10 =
ZR
I11 =
ZR
I20 =
ZR
I21 =
ZR
I22 =
R
(4)
w4 (x) |f (x)|2 dx,
(2)
w4 (x) |f (x)|2 dx,
2
(2)
w4 (x) |f 0 (x)| dx,
w4 (x) |f (x)|2 dx,
2
w4 (x) |f 0 (x)| dx,
2
w4 (x) |f 00 (x)| dx,
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and
Z
(2)
w4
I101 =
(x) Re if
(x) f 0
Z
(2)
(x) dx = − w4 (x) Im f (x) f 0 (x) dx,
R
R
Z
Z
w4 (x) Im f (x) f 0 (x) dx,
I201 =
w4 (x) Re −if (x) f 0 (x) dx =
R
R
Z
Z
I202 =
w4 (x) Re −f (x) f 00 (x) dx = − w4 (x) Re f (x) f 00 (x) dx,
R
R
Z
Z
I212 =
w4 (x) Re if 0 (x) f 00 (x) dx = − w4 (x) Im f 0 (x) f 00 (x) dx,
R
R
with w4 (x) = w(x)(x − xm )4 .
It is clear that (8.24) is equivalent to
p
8
(8.25)
M4 ≥
1
√
2π 4 2
John Michael Rassias
Title Page
q
4
|E4,f |,
where
Z h
On the Heisenberg-Pauli-Weyl
Inequality
(4)
(2)
E4,f =
w4 − 4β 2 w4 + 2β 4 w4 |f |2
R
2
2
(2)
(2)
+ 4 −w4 + 2β 2 w4 |f 0 | + 2w4 |f 00 | − 8β w4 − β 2 w4 Im f f 0
−4β 2 w4 Re f f 00 + 8βw4 Im f 0 f 00 (x) dx,
√
8
1
if |E4,f | < ∞ holds.We note that if |E4,f
|
=
,
then
from
(8.25)
we
find
M4 ≥
8
√
8
1
1√
∗
∗
(= H ), while if |E4,f | = 1, then M4 ≥ 2π 4 2 (> H ). Thus we observe
4π
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√
that the lower bound of 8 M4 is greater than H ∗ if |E4,f | > 18 ; the same with
H ∗ if |E4,f | = 81 ; and smaller than H ∗ , if 0 < |E4,f | < 81 . Finally, the above
inequality (8.24) generalizes (H4 ) of our Section 8 (there w = 1).
8.5.
First form of (8.3), if w = 1, xm = 0 and ξm = 0
(p)
We note that β = 2πξm = 0, wp (x) = xp , and wp (x) = p! (p = 1, 2, 3, 4, . . .).
Therefore the above-mentioned four special cases (i) – (iv) yield the four formulas:
Z
(8.26)
E1,f = − |f (x)|2 dx = −E|f |2 ,
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
R
Title Page
(8.27)
E2,f = 2
Z h
2
i
|f (x)|2 − x2 |f 0 (x)| dx,
Contents
R
(8.28)
E3,f = −3
Z h
2
2
2
0
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J
i
2 |f (x)| − 3x |f (x)| dx,
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and
(8.29)
E4,f = 2
Z h
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2
2
0
2
4
00
2
i
12 |f (x)| − 24x |f (x)| + x |f (x)| dx,
Page 84 of 151
R
respectively, if |Ep,f | < ∞ holds for p = 1, 2, 3, 4.
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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It is clear that, in general,
2
2
q
q
0
Aql =
β = 1, if l = q, and Aql =
β 2(q−l) , if l 6= q,
q
l
for 0 ≤ l ≤ q.
Thus, if β = 0, one gets

 1, if l = q,
(8.30) Aql =
= δlq (= the Kronecker delta), 0 ≤ l ≤ q.

0, if l 6= q
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
It is obvious, if β = 0, that
(8.31)
q−k
Bqkj = (−1)
q
k
q
j
β 2q−j−k = 0,
0 ≤ k < j ≤ q,
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such that j + k < 2q for 0 ≤ k < j ≤ q; that is, β 2q−j−k 6= β 0 (= 1) for
0 ≤ k < j ≤ q.
Therefore from (8.30) and (8.31) we obtain
Z
2
p−2q
wp(p−2q) (x) f (q) (x) dx,
(8.32)
Dq = Aqq Iqq = Iqq = (−1)
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if |Dq | < ∞, holds for 0 ≤ q ≤ p2 .
We note that if w = 1 and xm = 0, and ξm = 0 or β = 0, then wp (x) = xp
(p = 1, 2, 3, 4, . . .), and
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J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
wp(p−2q) (x) = (xp )(p−2q) = p (p − 1) · · · (p − (p − 2q) + 1) xp−(p−2q) ,
http://jipam.vu.edu.au
or
(8.33)
wp(p−2q)
p!
p! 2q
(x) =
x2q =
x ,
(p − (p − 2q))!
(2q)!
0≤q≤
hpi
2
.
From (8.32) and (8.33) we get the formula
Z
2
p−2q p!
(8.34)
Dq = (−1)
x2q f (q) (x) dx,
(2q)! R
p
if |Dq | < ∞ holds for 0 ≤ q ≤ 2 .
Therefore from (8.34) one finds that
John Michael Rassias
[p/2]
Ep,f =
X
Cq Dq
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q=0
[p/2] =
X
q=0
p
(−1)
p−q
q
p−q
q
Z
2
p−2q p!
2q (q)
(−1)
x f (x) dx ,
(2q)! R
or the formula
(8.35)
On the Heisenberg-Pauli-Weyl
Inequality
Ep,f =
Z [p/2]
X
p−q
(−1)
R q=0
p
p!
p − q (2q)!
p−q
q
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2
x f (q) (x) dx,
2q
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if |Ep,f | < ∞ holds for 0 ≤ q ≤
Let
p
, when w = 1 and xm = ξm = 0.
2
Z
(8.36)
(m2p )|f |2 =
R
x2p |f (x)|2 dx
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J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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be the 2pth moment of x for |f |2 about the origin xm = 0, and
Z
2
(8.37)
(m2p ) fˆ 2 =
ξ 2p fˆ (ξ) dξ
| |
R
2
the 2pth moment of ξ for fˆ about the origin ξm = 0. Denote
hpi
p!
p
p−q
p−q
, if p ∈ N and 0 ≤ q ≤
.
(8.38) εp,q = (−1)
q
p − q (2q)!
2
Thus from (8.35) and (8.38) we find
Z [p/2]
X
2
(8.39)
Ep,f =
εp,q x2q f (q) (x) dx,
R q=0
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Title Page
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 .
If w = 1 and xm = ξm = 0, one gets from (8.3) and (8.35) – (8.38) the
following Corollary 8.2.
Corollary 8.2. Assume that f : R → C is a complex valued function of a real
variable x, w = 1, xm = ξm = 0, and fˆ is the Fourier transform of f , described
in Theorem 8.1. Denote (m2p )|f |2 or fˆ 2 and εp,q as in (8.36) (or (8.37)) and
| |
(8.38), respectively for all p ∈ N.
If f ∈ L2 (R), and all the above assumptions hold, then
v
u
u[p/2]
q
X
q
1 u
p 2p
t
√
(8.40)
(m2p )|f |2 2p (m2p ) fˆ 2 ≥
εp,q (m2q ) f (q) 2 ,
p
| |
| |
2π 2 q=0
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holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 , where
Z
2
(8.41)
(m2q ) f (q) 2 =
x2q f (q) (x) dx.
| |
R
Equality in (8.40) holds iff the differential equation f (p) (x) = −2cp xp f (x) of
pth order holds for some cp > 0, and any fixed but arbitrary p ∈ N.
If q = 0, then we note that (8.41) yields
Z
(m0 )|f |2 =
|f (x)|2 dx = E|f |2 .
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
R
We also note that if p = 5, then [p /2] = 2; q = 0, 1, 2. Thus from (8.39) we get
5
5!
5−0
5−0
ε5,0 = (−1)
0
5 − 0 (2 · 0)!
ε5,1 = (−1)
5!
5
5 − 1 (2 · 1)!
5!
5
5 − 2 (2 · 2)!
5−1
1
5−2
2
= 300,
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and
5−2
ε5,2 = (−1)
= −25.
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Therefore
(8.42)
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JJ
J
= −120,
5−1
Title Page
E5,f = −5
Z h
R
2
2
0
2
4
00
2
i
24 |f (x)| − 60x |f (x)| + 5x |f (x)| dx,
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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if |E5,f | < ∞ holds.
Similarly if p = 6, then p2 = 3; q = 0, 1, 2, 3. Thus from (8.39) one finds
ε6,1 = −2160,
ε6,0 = 720,
and ε6,3 = −2.
ε6,2 = 270,
Therefore
(8.43) E6,f = 2
Z h
2
360 |f (x)|2 − 1080x2 |f 0 (x)|
R
2
2
+135x4 |f 00 (x)| − x6 |f 000 (x)|
i
dx,
if |E6,f | < ∞ holds. In the same way one gets
Z h
2
(8.44) E7,f = −7
720 |f (x)|2 − 2520x2 |f 0 (x)|
John Michael Rassias
Title Page
R
2
2
+420x4 |f 00 (x)| − 7x6 |f 000 (x)|
(8.45) E8,f = 2
Z h
R
On the Heisenberg-Pauli-Weyl
Inequality
2
i
dx,
2
20160 |f (x)|2 − 80640x2 |f 0 (x)| + 16800x4 |f 00 (x)|
2 i
2
6
000
8 (4)
−448x |f (x)| + x f (x) dx,
and
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(8.46) E9,f = −9
Z h
2
40320 |f (x)|2 − 181440x2 |f 0 (x)|
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R
2 i
2
2
+45360x4 |f 00 (x)| − 1680x6 |f 000 (x)| + 9x8 f (4) (x) dx,
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if |Ep,f | < ∞ holds for p = 7, 8, 9. We note that the cases Ep,f : p = 1, 2, 3, 4
are given above via the four formulas (8.26) – (8.29).
8.6.
Second form of (8.3), if ξm = 0
In general for wp (x) = w(x)xpδ with xδ = x − xm , where xm is any fixed and
arbitrary real, and w : R → R a real valued weight function, as well as, ξm = 0,
we get from (5.1) and (8.32) that
Z
2
p−2q
Dq = Iqq = (−1)
(w (x) xpδ )(p−2q) f (q) (x) dx
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
R
Z p−2q
X p − 2q 2
p−2q
= (−1)
w(m) (x) (xpδ )(p−2q−m) f (q) (x) dx
m
R m=0
Z p−2q
X p − 2q p!
p−2q
= (−1)
w(m) (x)
m
(p − (p − 2q − m))!
R m=0
2
p−(p−2q−m) (q)
× xδ
f (x) dx,
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or
Z "p−2q
X
(8.47) Dq = (−1)p−2q
R
m=0
p!
(2q + m)!
p − 2q
m
#
w(m) (x) xm
δ
(q)
2
× x2q
(x) dx,
δ f
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if |Dq | < ∞ holds for 0 ≤ q ≤
p
.
2
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If m = 0, then one finds from (8.47) the formula (8.34). Therefore from
(8.47) one gets that
Z [p/2]
X
p
p−q
(8.48) Ep,f =
(−1)
q
p−q
R q=0
#
"p−2q
X
(q)
p!
p − 2q
f (x)2 dx,
×
w(m) (x) xm
x2q
δ
δ
m
(2q + m)!
m=0
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 , when w : R → R is a real valued weight
function, xm any fixed but arbitrary real constant and ξm = 0. If m = 0 and
xm = 0, then we find from (8.48) the formula (8.35).
If we denote
p
p−q
p−q
(8.49) εp,q,w (x) = (−1)
q
p−q
"p−2q
#
X
p!
p − 2q
×
w(m) (x) xm
,
δ
m
(2q + m)!
m=0
p−q
Ep,f =
Z [p/2]
X
δ
[p/2] Z
X
q=0
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2
εp,q,w (x)x2q f (q) (x) dx
R q=0
=
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then one gets from (8.48) that
(8.50)
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Inequality
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(q)
2
εp,q,w (x) x2q
(x) dx,
δ f
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if |Ep,f | < ∞ holds for 0 ≤ q ≤
It is clear that the formula
[p/2]
(8.51) Ep,f =
X
p−q
(−1)
q=0
×
Z "p−2q
X
R
m=0
p
.
2
p
p−q
p!
(2q + m)!
p−q
q
p − 2q
m
#
w(m) (x) xm
δ
(q)
2
× x2q
(x) dx,
δ f
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 .
Therefore from (8.3) and (8.49) – (8.50) we get the following Corollary 8.3.
Corollary 8.3. Assume that f : R → C is a complex valued function of a real
variable x, w : R → R a real valued weight function, xm any fixed but arbitrary
real number, ξm = 0, and fˆ is the Fourier transform of f , described in our above
theorem. Denote (µ2p )w,|f |2 , (m2p ) fˆ 2 and εp,q,w (x) as in the preliminaries of
| |
the above theorem, (8.37) and (8.49), respectively, for all p ∈ N.
If f ∈ L2 (R), and all the above assumptions hold, then
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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(8.52)
q
q
2p
(µ2p )w,|f |2 2p (m2p ) fˆ 2
| |
 1

[p/2]
p
Z
2 1 X
2q (q)
√
εp,q,w (x) xδ f (x) dx ,
≥
2π p 2 q=0 R
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holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 .
Equality in (8.52) holds iff the differential equation
f (p) (x) = −2cp xpδ f (x)
of pth order holds for some cp > 0, and any fixed but arbitrary p ∈ N.
We note that for p = 2; q = 0, 1 and w2 (x) = w (x) x2δ , with xδ = x − xm ;
ξm = 0, we get from (8.49) that
ε2,0,w (x) = 2w (x) + 4w0 (x) xδ + w00 (x) x2δ ,
On the Heisenberg-Pauli-Weyl
Inequality
and
John Michael Rassias
ε2,1,w (x) = −2w (x) .
Therefore from (8.48) one obtains
Z
(8.53) E2,f =
2w (x) + 4w0 (x) xδ + w00 (x) x2δ |f (x)|2
R
i
2
−2w (x) x2δ |f 0 (x)| dx, if |E2,f | < ∞.
This result (8.53) can be found also from (8.20), where β = 2πξm = 0 and thus
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(8.54)
E2,f =
Z h
2
(2)
w2 (x) |f (x)|2 − 2w2 (x) |f 0 (x)| dx,
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R
if |E2,f | < ∞ holds, with
Page 93 of 151
2
w2 (x) = w (x) (x − xm ) =
(1)
i
w (x) x2δ ,
w2 (x) = 2w (x) xδ + w0 (x) x2δ
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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and
(2)
w2 (x) = 2w (x) + 4w0 (x) xδ + w00 (x) x2δ .
8.7.
Third form of (8.3), if w = 1
In general with any fixed but arbitrary real numbers xm , ξm , xδ = x − xm and
ξδ = ξ − ξm , one finds
wp = xpδ ,
wp(r) = (xpδ )(r)
On the Heisenberg-Pauli-Weyl
Inequality
p!
xp−r ,
=
(p − r)! δ
and
wp(p−2q) =
John Michael Rassias
p! 2q
x .
(2q)! δ
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Therefore the integrals Iql , Iqkj of the above theorem take the form
(8.55)
Iql = (−1)p−2q
p!
(µ2q ) f (l) 2 ,
| |
(2q)!
0≤l≤q≤
hpi
2
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and
(8.56)
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p−2q
Iqkj = (−1)
p!
(µ2q )fkj ,
(2q)!
0≤k<j≤q≤
hpi
2
,
where fkj : R → R are real valued functions of x, such that
hpi
,
(8.57)
fkj (x) = Re rqkj f (k) (x) f (j) (x) , 0 ≤ k < j ≤ q ≤
2
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with rqkj = (−1)q−
k+j
2
, and
Z
(8.58)
(µ2q ) f (l) 2 =
| |
(l)
2
x2q
(x) dx,
δ f
0≤l≤q≤
hpi
R
2
is the 2q th moment of x for f (l) , and
Z
(8.59)
(µ2q )fkj =
x2q
δ fkj (x)dx,
0≤k<j≤q≤
hpi
R
2
2
,
,
is the 2q th moment of x for fkj .
We note that if 0 ≤ k = j = l ≤ q ≤ p2 , then
(8.60)
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
2
fll (x) = sql f (l) (x) ,
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and thus
(8.61)
hpi
(µ2q )fll = sql (µ2q ) f (l) 2 , 0 ≤ l ≤ q ≤
,
| |
2
where sql = (−1)q−l .
We consider Aql and Bqkj and β as in the theorem, and εp,q as in (8.38).
Therefore
" q
#
X
X
p−2q p!
(8.62) Dq = (−1)
Aql (µ2q ) f (l) 2 + 2
Bqkj (µ2q )fkj ,
| |
(2q)! l=0
0≤k<j≤q
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if |Dq | < ∞ holds for 0 ≤ q ≤
p
. From (8.55) – (8.62) one gets
2
[p/2]
(8.63)
Ep,f =
" q
X X
q=0
#
Aql (µ2q ) f (l) 2 + 2
| |
l=0
X
Bqkj (µ2q )fkj ,
0≤k<j≤q
if |Ep,f | < ∞ holds, for any fixed but arbitrary p ∈ N.
If w = 1 one gets from (8.3) and (8.55) – (8.63) the following Corollary 8.4.
Corollary 8.4. Assume that f : R → C is a complex valued function of a real
variable x, w = 1, xm and ξm any fixed but arbitrary real numbers, xδ = x−xm
and ξδ = ξ − ξm , wp (x) = xpδ , and fˆ is the Fourier transform of f , described
in our theorem. Let
Z
2
(µ2p )|f |2 =
x2p
δ |f (x)| dx,
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
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and
Z
(µ2p ) fˆ 2 =
| |
R
2
ξδ2p fˆ(ξ) dξ
2
be the 2pth moment of x for |f |2 , and the 2pth moment of ξ for fˆ , respectively.
Denote (µ2q ) f (l) 2 , (µ2q )fkj (with fkj : R → R as in (8.57)) , εp,q and Aql , Bqkj
| |
via (8.58), (8.59), (8.38) and the preliminaries of the theorem, respectively for
all p ∈ N. Also denote
q
q
Up = 2p (µ2p )|f |2 2p (µ2p ) fˆ 2 .
| |
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If f ∈ L2 (R), and all the above assumptions hold, then

" q
[p/2]
X
X
(8.64)
Up ≥ Hp∗ 
εp,q
Aql (µ2q ) f (l) 2
| |
q=0
l=0
+2
X
0≤k<j≤q
## p1
Bqkj (µ2q )fkj √
holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 , where Hp∗ = 1/ 2π p 2
(for p ∈ N) is the generalized Heisenberg constant.
Equality in (8.64) holds iff the a-differential equation
fa(p) (x) = −2cp xpδ fa (x) , a = −2πξm i,
holds for some cp > 0, and any fixed but arbitrary p ∈ N.
We call Up the uncertainty product due to the Heisenberg uncertainty principle (8.64).
We note that if f : R → R is a real valued function of a real variable, in the
above Corollary 8.4, then
hpi
,
(8.65)
fkj = f (k) f (j) Re (rqkj ) , for 0 ≤ k < j ≤ q ≤
2
where rqkj ∈ {±1, ±i}, such that

if 2q ≡ (k + j) (mod 4)
 1,
; and
rqkj =

−1, if 2q ≡ (k + j + 2) (mod 4)
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John Michael Rassias
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rqkj =

 i,

if 2q ≡ (k + j + 1) (mod 4)
−i, if 2q ≡ (k + j + 3) (mod 4) .
Thus
(8.66) fkj = f (k) f (j)

1,
if 2q ≡ (k + j) (mod 4)





−1, if 2q ≡ (k + j + 2) (mod 4)
,
×





0,
if 2q ≡ (k + j + 1) (or (k + j + 3)) (mod 4)
for 0 ≤ k < j ≤ q ≤ p2 .
Therefore
(8.67)
(µ2q )fkj = (µ2q )f (k) f (j)

1,
if 2q ≡ (k + j) (mod 4)





−1, if 2q ≡ (k + j + 2) (mod 4)
×
,





0,
if 2q ≡ (k + j + 1) (or (k + j + 3)) (mod 4)
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where
Z
(µ2q )f (k) f (j) =
R
f (k) f (j) (x) dx,
x2q
δ
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for 0 ≤ k < j ≤ q ≤
p
.
2
Similarly if f (k) f (j) : R → R, for 0 ≤ k < j ≤ q ≤
functions of a real variable x, we get analogous results.
p
, are real valued
2
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Inequality
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9.
Gaussian Function
Consider w = 1, xm and ξm means and the Gaussian function f : R → C, such
2
that f (x) = c0 e−cx , where c0 , c are constants and c0 ∈ C, c > 0. It is easy to
prove the integral formula
Z
Γ p + 12
2p −2cx2
(9.1)
x e
dx =
1 , c > 0,
R
(2c)p+ 2
for all p ∈ N and p = 0, where Γ is the Euler gamma function [21], such that
1
1 · 3 · · · · · (2p − 1) √
π
Γ p+
=
2
2p
√
for p ∈ N and Γ 12 = π for p = 0. Note that the mean xm of x for |f |2 is
given by
Z
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John Michael Rassias
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2
x |f (x)| dx = 0.
xm =
R
Also from Gasquet et al [8, p. 159-161], by applying differential equations [25],
one gets that the Fourier transform fˆ : R → C is of the form
r
π − π2 ξ 2
(9.2)
fˆ (ξ) = c0
e c , c0 ∈ C, c > 0.
c
2
In this case the mean ξm of ξ for fˆ is given by
Z
ξm =
R
ˆ 2
ξ f (ξ) dξ = 0.
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Therefore from (8.36) – (8.37) with means xm = 0, ξm = 0 and from (9.1) –
(9.2) one finds that the 2pth power of the left-hand side of the inequality (8.40)
of the Corollary 8.2 is
(9.3) (m2p )|f |2 · (m2p ) fˆ 2
| |
Z
Z
2 2
2p
2p ˆ
=
x |f (x)| dx
ξ f (ξ) dξ
R
R
Z
Z
π2
π
2
4
2p −2cx
2p −2c∗ ξ 2
∗
= |c0 |
x e
dx
ξ e
dξ
where c =
c
c
R
R
4
4
1
1
1 |c0 |
|c0 | Γ p + 2 Γ p + 2
∗ 2p
2Γ2 p +
,
=π
1 = Hp
1
p+
p+
2
c
c (2c) 2 (2c∗ ) 2
for all fixed but arbitrary p ∈ N, c0 ∈ C, and c > 0 where Hp∗ = 2π1√
. We
p
2
note that
Z
(9.4)
E|f |2 =
|f (x)|2 dx
R Z
2
= |c0 |2 e−2cx dx
R
|c0 |2
1
=√ Γ
2
2c
r
√
π
1
2
= |c0 |
, where Γ
= π.
2c
2
If we denote
(2p − 1)!! = 1 · 3 · 5 · · · · · (2p − 1) , 0!! = (−1)!! = 1,
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Inequality
John Michael Rassias
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for p ∈ N and p = 0, respectively, then
1
(2p − 1)!! √
=
(9.5)
Γ p+
π for p ∈ N, and
2
2p
√
1
Γ
= π for p = 0.
2
We consider the Legendre duplication formula for Γ ([18], [21])
1
22p−1
, p∈N
(9.6)
Γ (2p) = √ Γ (p) Γ p +
2
π
and the factorial formula
(9.7)
Γ (p + 1) = p!, p ∈ N0 = N ∪ {0}.
We take the Hermite polynomial ([18], [21])
q (q − 1)
(2x)q−2
1!
q (q − 1) (q − 2) (q − 3)
+
(2x)q−4
2!
q
q
q!
q ∈ N0 ,
− · · · + (−1)[ 2 ] q (2x)q−2[ 2 ] ,
!
2
where 2q = 2q if q is even and 2q = q−1
if q is odd. We consider the Rodrigues
2
formula ([18], [21])
q 2 d
−x2
(9.9)
Hq (x) = (−1)q ex
e
, q ∈ N0 .
dxq
(9.8) Hq (x) = (2x)q −
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John Michael Rassias
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If one places
√
cx on x into (9.8) and employs
d
dx
1 d
d
√
(·) =
(·) √
=√
(·) ,
dx
d ( cx)
d ( cx)
c dx
(9.10)
then he proves the generalized Rodrigues formula
q √ q − q cx2 d
−cx2
2
(9.11)
Hq
cx = (−1) c e
e
, c > 0, q ∈ N0 .
dxq
In this paper we have 0 ≤ q ≤ p2 , p ∈ N. From (9.11) with f (x) =
2
c0 e−cx , c0 ∈ C, c > 0, we get
√ q
dq
f (x) = (−1)q c 2 f (x) Hq
cx ,
q
dx
(9.12)
Z
2
x2q f (q) (x) dx
ZR
√ 2
q
=
x2q (−1)q c 2 f (x) Hq
cx dx
R
Z
√ 2
2 2 q
= |c0 | c
x2q e−2cx Hq
cx dx.
(m2q ) f (q) 2 =
| |
R
Substituting y =
(9.14)
√
John Michael Rassias
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and thus the moment
(9.13)
On the Heisenberg-Pauli-Weyl
Inequality
cx, c > 0 into (9.13) one gets
Z
|c0 |2
2
(m2q ) f (q) 2 = √
y 2q e−2y Hq2 (y) dy.
| |
c R
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We consider the Hermite polynomial
[ 2q ]
X
hpi
q!
q−2k
,
Hq (y) =
(−1)
(2y)
, 0≤q≤
k! (q − 2k)!
2
k=0
(9.15)
k
and the Lagrange identity of the second form (7.3). Setting
q!
q
k
k (2k)!
q−2k
rk = (−1)
2
= (−1)
2q−2k
2k
k! (q − 2k)!
k!
On the Heisenberg-Pauli-Weyl
Inequality
with
(2k)!
k!
q
2k
=
q!
,
k! (q − 2k)!
John Michael Rassias
and denoting
Title Page
A∗qk =
(2k)!
k!
2 q
2k
2
22(q−2k) ∈ R,
Contents
JJ
J
∗
rqkj
= 4q−(k+j) ∈ R,
s∗qkj = (−1)k+j ∈ R,
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and
∗
Bqkj
= s∗qkj
(2k)! (2j)!
k!j!
q
2k
q
2j
∈ R,
∗
∗
one gets that rk2 = A∗qk and rk rj = rqkj
Bqkj
.
Thus employing (9.15) and substituting zk = y q−2k in (7.3) we find
zk zj = y 2(q−k−j) , rk zk = (−1)k
q!
(2y)q−2k ,
k! (q − 2k)!
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and

Hq2 (y) = 
(9.16)
[q/2]
X
2
rk zk 
k=0
=
[q/2]
X
A∗qk y 2q−4k + 2
k=0
X
∗
∗
rqkj
Bqkj
y 2(q−k−j) ,
0≤k<j≤[ 2q ]
On the Heisenberg-Pauli-Weyl
Inequality
for 0 ≤ q ≤ p2 , p ∈ N. Let us denote
Z
Z
2
∗
4(q−k) −2y 2
∗
∗
Jqk =
y
e
dy, Jqkj =
y 2(2q−k−j) e−2y dy, Ãqk = A∗qk Jqk
,
R
John Michael Rassias
R
∗
∗
∗
and B̃qkj = rqkj
Bqkj
Jqkj
. Therefore from (9.14) and (9.16) one gets


[q/2]
2
hpi
X
X
|c0 |
.
(9.17) (m2q ) f (q) 2 = √ 
Ãqk + 2
B̃qkj  , 0 ≤ q ≤
| |
2
c k=0
0≤k<j≤[q/2]
(9.18)
(4 (q − k) − 1)!!
=
16q−k
r
π
(2 (2q − k − j) − 1)!!
∗
, and Jqkj
=
2
42q−k−j
(9.19)
Ãqk
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.
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From (9.18) one gets
1
= 2q
2
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From (9.1) and (9.5) we find
∗
Jqk
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r 2 2
π
(2k)!
q
(4q − 4k − 1)!!,
2 2k
k!
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r π
q
q
(9.20) B̃qkj = (−1)
2j
2 2k
(2k)! (2j)!
×
(4q − 2k − 2j − 1)!!,
k!j!
0 ≤ k < j ≤ 2q ,0 ≤ q ≤ p2 , p ∈ N. From (9.5) one finds
1
22q−2k
,
(9.21)
(4q − 4k − 1)!! = √ Γ 2q − 2k +
2
π
k+j
(9.22)
1
22q
1
22q−k−j
.
(4q − 2k − 2j − 1)!! = √ Γ 2q − k − j +
2
π
1
(2p)!
22p
= √ Γ p+
2
p!
π
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,
p ∈ N.
Therefore from (9.19) – (9.22) and placing k, j on p into (9.23) we find
2 1 2k
1
1
q
2
(9.24)
Ãqk = √ 2
Γ k+
Γ 2q − 2k +
,
2k
2
2
π 2
(9.25) B̃qkj
John Michael Rassias
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Also from (9.6) – (9.7) we get
(9.23)
On the Heisenberg-Pauli-Weyl
Inequality
1
q
q
k+j k+j
= √ (−1) 2
2k
2j
π 2
1
1
1
×Γ k+
Γ j+
Γ 2q − k − j +
,
2
2
2
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for all 0 ≤ k < j ≤
[q/2]
X
q
, 0 ≤ q ≤ p2 , p ∈ N. Let us denote
2
2
1
2
Γ
(9.26) Γq =
Γ 2q − 2k +
2
k=0
X
q
q
k+j k+j
+2
(−1) 2
2k
2j
0≤k<j≤[q/2]
1
1
1
Γ j+
Γ 2q − k − j +
.
×Γ k+
2
2
2
2k
q
2k
2
1
k+
2
John Michael Rassias
From (9.26) one gets
(9.27)
[q/2]
X
k=0
Ãqk + 2
1
B̃qkj = √ Γq .
π 2
0≤k<j≤[q/2]
X
Thus from (9.17) and (9.27) we find
(9.28)
On the Heisenberg-Pauli-Weyl
Inequality
|c0 |2 1
(m2q ) f (q) 2 = √ √ Γq ,
| |
c π 2
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for all 0 ≤ q ≤
(9.29)
p
, p ∈ N, and c0 ∈ C, c > 0. Let us denote
2
[p/2]
X
Γ∗p = εp,q Γq ,
q=0
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where εp,q is given as in (8.38). Therefore from the 2pth power of the right-hand
side of the inequality (8.40) of the Corollary 8.2 and (9.28) – (9.29) we get
2

[p/2]
(9.30)
Hp∗
2p
X

q=0
εp,q (m2q ) f (q) 2  = Hp∗
| |
4
2p 1
∗ 2 |c0 |
Γ
,
2π 2 p
c
for all p ∈ N, c0 ∈ C, and c > 0 where Hp∗ = 2π1√
.
p
2
2
If fˆ : R → C is the Fourier transform of f of the form f (x) = c0 e−cx
(c0 ∈ C, c > 0),
as in the abstract, xm the mean of x for |f |2 , and ξm the
given
2
mean of ξ for fˆ , then
(9.31)
E|f |2 ∗
|c0 |2 1 ∗
|Ep,f | = √ √ Γp = √ Γp ,
π π
c π 2
for any fixed but arbitrary p ∈ N. For instance, if c0 = 1 and c = 12 , then
Γ∗p = π |Ep,f | ,
p ∈ N.
Therefore from (8.40), (9.3) and (9.30) we get the following Corollary 9.1.
Corollary 9.1. Assume that Γ is the Euler gamma function defined by the formula [21]
Z ∞
(9.32)
Γ (z) =
tz−1 e−t dt, Re(z) > 0,
0
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√
whenever the complex variable z = Re (z) + i Im (z), i = −1, has a positive real part Re (z). Denote εp,q , Γq and Γ∗q as in (8.38), (9.26) and (9.29),
respectively. Let us consider the non-negative real function R : N → R, such
that
Γ p + 21
R (p) =
, p ∈ N = {1, 2, 3, . . .}.
Γ∗p
Then the extremum principle
1
(9.33)
R (p) ≥
,
2π
holds for any fixed but arbitrary p ∈ N. Equality holds for p = 1.
For instance, if p ∈ N9 = {1, 2, 3, . . ., 9}, then
429 1
1
≤ R (p) ≤
·
.
2π
23 2π
9.1.
i) If p = 1, then q √= 0. Thus Γ0 = Γ3
Γ∗1 = |ε1,0 Γ0 | = π π.
√
But Γ 1 + 21 = 21 π. Hence R (1) =
(9.33) holds for p = 1.
2
1
.
2π
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First nine cases of (9.33)
1
On the Heisenberg-Pauli-Weyl
Inequality
√
= π π, ε1,0 = −1, and
Therefore the equality in
ii) If p = 2, then q = 0, 1.Thus
√
Γ0 = π π, ε2,0 = 2;
1
1
3 √
2
Γ1 = Γ
Γ 2+
= π π,
2
2
4
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ε2,1 = −2,
and
1 √
Γ∗2 = |ε2,0 Γ0 + ε2,1 Γ1 | = π π.
2
3√
1
1
But Γ 2 + 2 = 4 π. Hence R (2) = 3 · 2π .
Therefore the inequality in (9.33) holds for p = 2.
√
√
iii) If p = 3, then q = 0, 1.Thus Γ0 = π π, ε3,0 = −6; Γ1 = 34 π π, ε3,1 = 9,
and
3 √
Γ∗3 = |ε3,0 Γ0 + ε3,1 Γ1 | = π π.
4
15 √
1
1
But Γ 3 + 2 = 8 π. Hence R (3) = 5 · 2π .
Therefore the inequality in (9.33) holds for p = 3.
√
iv) If p = 4, then q = 0, 1, 2. Thus Γ0 = π π, ε4,0 = 24; Γ1 =
ε4,1 = −48; ε4,2 = 2,
1
X
2
1
Γ2 =
2
Γ
Γ 4 − 2k +
2
k=0
X
2
2
+2
(−1)k+j 2k+j
2k
2j
0≤k<j≤1
1
1
1
×Γ k +
Γ j+
Γ 4−k−j+
2
2
2
2k
2
2k
2
1
k+
2
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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3 √
π π,
4
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1
1
1
1
2 2
=Γ
Γ 4+
+2 Γ 1+
Γ 2+
2
2
2
2
1
1
1
+ 2 (−1) 2Γ
Γ 1+
Γ 3+
2
2
2
105 + 12 − 60 √
57 √
=
π π = π π,
16
16
2
because (k, j) ∈ {(0, 1)}, and
On the Heisenberg-Pauli-Weyl
Inequality
Γ∗4 = |ε4,0 Γ0 + ε4,1 Γ1 + ε4,2 Γ2 |
57 √
39 √
3
π π = π π.
= 24 + (−48) + 2
4
16
8
√
π. Hence R (4) = 35
But Γ 4 + 21 = 105
· 1.
16
13 2π
John Michael Rassias
Title Page
Therefore the inequality in (9.33) holds for p = 4.
√
1
945 √
v) If p = 5, then q = 0, 1, 2. Thus Γ∗5 = 255
π
π.
But
Γ
5
+
=
π.
16
2
32
1
Hence R (5) = 63
·
.
Therefore
the
inequality
in
(9.33)
holds
for
p
=
5.
17 2π
√
1
vi) If p √
= 6, then q = 0, 1, 2, 3. Thus Γ∗6 = 855
π
π.
But
Γ
6
+
=
32
2
10395
231
1
π.
Hence
R
(6)
=
·
.
Therefore
the
inequality
in
(9.33)
holds
64
19
2π
for p = 6.
√
vii) If p =√ 7, then q = 0, 1, 2, 3. Thus Γ∗7 = 7245
π π. But Γ 7 + 2 =
64
135135
429 1
π. Hence R (7) = 23 · 2π . Therefore the inequality in (9.33) holds
128
for p = 7.
1
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√
viii) If p = √
8, then q = 0, 1, 2, 3, 4. Thus Γ∗8 = 192465
π π. But Γ 8 + 12 =
128
2027025
1
π. Hence R (8) = 495
· 2π
. Therefore the inequality in (9.33)
256
47
holds for p = 8.
ix) If p = 9,
then q = 0, 1, 2, 3, 4. Thus Γ∗9 =
√
34459425
π.
512
Hence R (9) =
p = 9.
In fact ,
ε9,0 = −9!,
12155
827
ε9,1
·
1
.
2π
2344545 √
π π.
256
But Γ 9 +
1
2
=
Therefore the inequality in (9.33) holds for
On the Heisenberg-Pauli-Weyl
Inequality
9!
=9· ,
2!
ε9,2
9 9!
=−
7 4!
7
2
,
ε9,3
9 9!
=
6 6!
6
3
John Michael Rassias
,
ε9,4 = −81
and
Γ∗9 = |ε9,0 Γ0 + ε9,1 Γ1 + ε9,2 Γ2 + ε9,3 Γ3 + ε9,4 Γ4 | ,
√
√
√
where Γ0 = π π, Γ1 = 34 π π, and Γ2 = 57
π π from the above case iv).
16
Besides
2 1
X
1
1
3
2k
2
Γ3 =
2
Γ k+
Γ 6 − 2k +
2k
2
2
k=0
X
3
3
+2
(−1)k+j 2k+j
2k
2j
0≤k<j≤1
1
1
1
×Γ k +
Γ j+
Γ 6−k−j+
2
2
2
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2 1
1
1
1
3
2
2
=Γ
Γ 6+
+2
Γ 4+
Γ 1+
2
2
2
2
2
1
1
1
3
3
+ 2 (−1) 2
Γ
Γ 1+
Γ 5+
0
2
2
2
2
2835 √
=
π π,
64
2
because (k, j) ∈ {(0, 1)}, and
2
X
2
1
1
4
2k
2
Γ4 =
2
Γ k+
Γ 8 − 2k +
2k
2
2
k=0
X
4
4
+2
(−1)k+j 2k+j
2k
2j
0≤k<j≤2
1
1
1
×Γ k +
Γ j+
Γ 8−k−j+
2
2
2
√
273105
=
π π,
256
because (k, j) ∈ {(0, 1), (0, 2), (1, 2)}.We note that if one denotes R∗ (p) =
2πR (p), then he easily gets R∗ (p) ≥ 1 for any p ∈ N.
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
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Corollary 9.2. Assume that Γ is defined by (9.32). Consider the Gaussian f :
2
R → C such that f (x) = c0 e−cx , where c0 , c are fixed but arbitrary constants
and c0 ∈ C, c > 0. Assume that xm is the mean of x for |f |2 . Consider fˆ : R
→ C the Fourier transform of f , given as in the abstract and ξm the mean of ξ
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2
2
for fˆ . Denote (m2q ) f (q) 2 , the 2q th moment of x for f (q) about the origin,
| |
as in (8.41), and the real constants εp,q as in (8.38). Denote
Z
E|f |2 =
|f (x)|2 dx
R
and
[p/2]
(9.34)
Ep,f =
X
q=0
εp,q (m2q ) f (q) 2 ,
| |
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
p
if |Ep,f | < ∞ holds for 0 ≤ q ≤ 2 and any fixed but arbitrary p ∈ N.
Then the extremum principle
!
r
√
√ !
Γ p + 12
π ∗
1
c
π
(9.35) Rf (p) =
=
R (p) ≥
=
,
2
|Ep,f |
2E|f |2
2
2E|f |2
|c0 |
holds for any p ∈ N. Equality holds for p = 1.
For instance, if p ∈ N9 = {1, 2, 3, . . ., 9}, then Ep,f > 0 for p = 2, 3, 5, 8;
and < 0 for p = 1, 4, 6, 7, 9. Besides
r
r
c
c
1
429
1
≤ Rf (p) ≤
·
, if p ∈ N9 .
2
2
2
2
23 |c0 |
|c0 |
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The proof of Corollary 9.2 is a direct application of the above-mentioned
formula (9.31) and the Corollary 9.1 (or (9.33)). In fact,
1
|Ep,f |
Rf (p) = Γ p +
2
,
1
1 ∗ |c0 |2
√ Γp √
=Γ p+
2
c
π 2
√
√
c
= π 2 2 R (p)
|c0 |
√
r
2c 1
1
c
≥π
=
.
2 ·
2
2
|c0 | 2π
|c0 |
Besides from (8.26) one gets
Z
E1,f = − |f (x)|2 dx
R Z
2
= − |c0 |2 e−2cx dx
R
2
|c0 | √
|c0 |2
1
= −√
π = − √ 2Γ 1 +
,
2
2c
2c
or Rf (1) =
1
|c0 |2
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John Michael Rassias
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pc
, completing the proof of Corollary 9.2. We note that if
2
1 2
c0 = 1, c = 21 , or f (x) = e− 2 x , then Rf (p) ≥ 12 .
Also we note that the formula (9.28) is an interesting formula on moments
for Gaussians.
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9.2.
First nine cases of (9.35)
i) If p = 1, then
Z
2
|c0 |2 √
2
e−2cx dx = √
π.
2c
R
√
= −E|f |2 . But Γ 1 + 12 = 12 π. Hence
2
Z
|f (x)| dx = |c0 |
E|f |2 =
R
Thus from (8.26) we get E1,f
1
Rf (1) = Γ 1 +
2
1
|E1,f | =
|c0 |2
r
c √ .
= π 2E|f |2 .
2
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Therefore the equality in (9.35) holds for p = 1.
We note from (9.28) and (9.31) that q = 0 such that
√
√
√
2c
2c
2
Γ0 = π
2 (m0 )|f |2 = π
2 E|f | = π π,
|c0 |
|c0 |
and
√
√
√
2c
2c
2
Γ∗1 = π
2 |E1,f | = π
2 E|f | = Γ0 = π π,
|c0 |
|c0 |
respectively.
ii) If p = 2, then from (8.27) we get
Z
3
1
2
2
0
E2,f = 2 E|f |2 − x |f (x)| dx = 2 E|f |2 − E|f |2 = E|f |2 .
4
2
R
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But Γ 2 +
1
2
=
3√
π.
4
Hence
r
1
c
1
Rf (2) = Γ 2 +
|E2,f | = 3 ·
.
2
2
2
|c0 |
Therefore the inequality in (9.35) holds √
for p = 2. We note from (9.28)
and (9.31) that q = 0, 1 such that Γ0 = π π as in the above case i),
√
√ Z
√
2c
2c
2c 3
3 √
2
2
Γ1 = π
x2 |f 0 (x)| dx = π
2 (m2 )|f 0 |2 = π
2
2 E|f | = π π,
4
|c0 |
|c0 | R
|c0 | 4
and
√
Γ∗2
√
2c
2c 1
1
1 √
2
=π
2 |E2,f | = π
2 E|f | = Γ0 = π π,
2
2
|c0 |
|c0 | 2
respectively.
iii) If p = 3, then from (8.28) we find
Z
2
2
0
E3,f = −3 2E|f |2 − 3 x |f (x)| dx
R
3
= −3 2E|f |2 − 3 · E|f |2
4
3
= E|f |2 .
4
√
But Γ 3 + 21 = 15
π. Hence
8
r
1
1
c
Rf (3) = Γ 3 +
|E3,f | = 5 ·
.
2
2
2
|c0 |
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Inequality
John Michael Rassias
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Therefore the inequality in (9.35) holds for p = 3.
iv) If p = 4, then from (8.29) one finds
Z
Z
2
2
2
0
4 00
E4,f = 2 12E|f |2 − 24 x |f (x)| dx + x f (x) dx
R
R
3 57
= 2 12 − 24 · +
E|f |2
4 16
39
= − E|f |2 < 0.
8
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Hence
Rf (4) =
35
1
·
13 |c0 |2
r
c
.
2
Therefore the inequality in (9.35) holds for p = 4.
v) If p = 5, then from (8.42) one gets
Z
Z
2
2
2
0
4 00
E5,f = −5 24E|f |2 − 60 x |f (x)| dx + 5 x f (x) dx
R
R
3
57
255
= −5 24 − 60 · + 5 ·
E|f |2 =
E 2.
4
16
16 |f |
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Hence
Rf (5) =
1
63
·
17 |c0 |2
r
c
.
2
Therefore the inequality in (9.35) holds for p = 5.
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vi) If p = 6, then from (8.43) we get
Z
2
E6,f = 2 360E|f |2 − 1080 x2 |f 0 (x)| dx
R
Z
Z
2 2
0
0
4 00
6
0
+ 135 x f (x) dx − x f (x) dx
R
R
57 2835
3
−
E|f |2
= 2 360 − 1080 · + 135 ·
4
16
64
855
=−
E 2 < 0.
32 |f |
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
Hence
231
1
Rf (6) =
·
19 |c0 |2
r
c
.
2
Therefore the inequality in (9.35) holds for p = 6.
vii) If p = 7, then from (8.44) one obtains
Z
2
E7,f = −7 720E|f |2 − 2520 x2 |f 0 (x)| dx
R
Z
Z
2 2
4 00
6 000
+ 420 x f (x) dx − 7 x f (x) dx
R
R
3
57
2835
= −7 720 − 2520 · + 420 ·
−7·
E|f |2
4
16
64
7245
=−
E 2 < 0.
64 |f |
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Hence
429
1
Rf (7) =
·
23 |c0 |2
r
c
.
2
Therefore the inequality in (9.35) holds for p = 7.
viii) If p = 8, then from (8.45) we obtain
Z
2
E8,f = 2 20160E|f |2 − 80640 x2 |f 0 (x)| dx
R
Z
2
0
+ 16800 x4 f 0 (x) dx
Z R
Z
2
2
6 000
8 (4)
−448 x f (x) dx + x f (x) dx
R
R
3
57
= 2 20160 − 80640 · + 16800 ·
4
16
2835 273105
−448 ·
+
E|f |2
64
256
192465
=
E|f |2 .
128
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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Hence
Rf (8) =
495
1
·
47 |c0 |2
r
c
.
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Therefore the inequality in (9.35) holds for p = 8.
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ix) If p = 9, then from (8.46) one finds
Z
2
E9,f = −9 40320E|f |2 − 181440 x2 |f 0 (x)| dx
R
Z
2
0
+ 45360 x4 f 0 (x) dx
Z R Z
2
2
6 000
8 (4)
−1680 x f (x) dx + 9 x f (x) dx
R
R
57
3
= 9 40320 − 181440 · + 45360 ·
4
16
2835
273105
−1680 ·
+9·
E|f |2
64
256
2344545
=−
E|f |2 < 0.
256
Hence
r
12155
c
1
Rf (9) =
·
.
2
2
827 |c0 |
Therefore the inequality in (9.35) holds for p = 9.
We note that, from the Corollary 9.1,
−2x2
f (x) = e
because |Ep,f | =
1 ∗
Γ,
2π p
from (9.31).
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Rf (p) = 2πR (p) = R∗ (p) ≥ 1
for any p ∈ N, if
On the Heisenberg-Pauli-Weyl
Inequality
√ π
=
,
2
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Corollary 9.3. Assume that the Euler gamma function Γ is defined by (9.32).
2
Consider the Gaussian function f : R → C of the form f (x) = c0 e−c(x−x0 ) ,
where c0 , c,x0 are fixed but arbitrary constants and c0 ∈ C, c > 0, x0 ∈ R.
Assume that the mathematical expectation E(x − x0 ) of x − x0 for |f |2 equals
to
Z
xm =
(x − x0 ) |f (x)|2 dx = 0.
R
Consider the Fourier transform fˆ : R → C of f , given as in the abstract of this
2
paper, and ξm the mean of ξ for fˆ . Denote by (m2q ) f (q) 2 the 2q th moment of
| |
(q) 2
x for f
about the origin, as in (8.41), and the constants εp,q as in (8.38).
Consider
[p/2]
X
Ep,f =
εp,q (m2q ) f (q) 2 ,
| |
q=0
if |Ep,f | < ∞ for 0 ≤ q ≤ p2 , and any fixed but arbitrary p ∈ N. If
2
d2p 2cx20
e
(> 0) denotes the 2pth order derivative of e2cx0 with respect to x0 ,
dx2p
0
then the extremum principle
√
4
(9.36)
|Ep,f | ≤
2
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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π
3p−1
2
1
· Γ2
p+
1
2
2
·
|c0 |
c
p+1
2
2
· e−cx0 ·
2p
d
2cx20
2p e
dx0
12
,
holds for any fixed but arbitrary p ∈ N. Equality holds for p = 1 and x0 = 0.
We note that xm = 0 even if x0 6= 0, while in the following Corollary 9.4 we
have xm = 0 only if x0 = 0.
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Proof. At first, we claim that the general integral formula
√
Z
π
1
d2p
2
2
2p −2c(x−x0 )2
(9.37)
x e
dx = 4p+ 1 · 2p+ 1 · e−2cx0 · 2p e2cx0 ,
dx0
2 2 c 2
R
c > 0, x0 ∈ R,
holds for all p ∈ N0 = N ∪ {0}.
We note that, if x0 = 0, then (9.1) follows. For example, if p = 1 and x0 = 0,
then
d d2 2cx20
2
2cx20
e
=
4cx
e
= 4c 1 + 4cx20 e2cx0 = 4c.
0
2
dx0
dx0
Thus (9.37) yields
r
r
Z
1
π
1
π
2 −2cx2
xe
dx =
· 1 · 4c =
.
2
2c
4c 2c
(4c)
R
This equals to
Γ(1+ 21 )
(2c)
1+ 1
2
because
√
1
1
π
3
= Γ
=
Γ
2
2
2
2
implying (9.1). A direct proof for this goes, as follows:
Z
Z
1
2 −2cx2
−2cx2
xe
dx =
xd e
−4c R
R
Z
1
−2cx2
−2cx2
=
xe
|R − e
dx
−4c
R
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John Michael Rassias
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1
=
4c
Z
2
e−2cx dx
R
Z
√ √
2
1
1 √
= √
e−( 2cx) d
π,
2cx = √
4c 2c R
4c 2c
2
because |x| e−2cx → 0, as |x| → ∞. It is easy to prove the integral formula
[21]
Z
√
√
2
(9.38)
xn e−(x−x0 ) dx = (2i)−n πHn (ix0 ) , i = −1, x0 ∈ R,
R
for all n ∈ N0 , where Hn is the Hermite polynomial ([18], [21]).
We note that if xm is the mean of x for |f |2 and x0 ∈ R − {0}, then
Z
xm =
x |f (x)|2 dx
R Z
2
= |c0 |2 xe−2c(x−x0 ) dx
R
Z
2 Z
2
|c0 |
2
−2c(x−x0 )2
=−
d e
+ x0 |c0 |
e−2c(x−x0 ) dx,
4c R
R
or
r
(9.39)
xm = x0 E|f |2 =
π |c0 |2
√ x0 ,
2 c
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because
Z
|f (x)|2 dx
R Z
2
= |c0 |2 e−2c(x−x0 ) dx
R
2 Z
√
√
2
|c0 |
=√
e−( 2c(x−x0 )) d
2c (x − x0 ) ,
2c R
E|f |2 =
On the Heisenberg-Pauli-Weyl
Inequality
or
r
E|f |2 =
(9.40)
π |c0 |2
√ .
2 c
On the other hand, if the mathematical expectation of x − x0 for |f |2 is
Z
E (x − x0 ) = xm =
(x − x0 ) |f (x)|2 dx,
R
then from (9.39) – (9.40) one gets
Z
Z
2
xm =
x |f (x)| dx − x0 |f (x)|2 dx = x0 E|f |2 − x0 E|f |2 ,
R
R
or
(9.41)
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xm = 0.
In this case the mathematical expectation xm is the mean of x for |f |2 only if
x0 = 0.
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√ We note that if one places q = 2p and ix0 i = −1 on x into (9.9) and
employs
d
d
dx0
d
(·) =
(·)
= −i
(·) ,
d (ix0 )
dx0
d (ix0 )
dx0
and thus
2p
d2p
p d
(·)
(·)
=
(−1)
dx2p
d (ix0 )2p
0
(9.42)
On the Heisenberg-Pauli-Weyl
Inequality
then he proves
(9.43)
2
H2p (ix0 ) = (−1)p e−x0
d2p x20
e ,
dx2p
0
p ∈ N0 .
Therefore from (9.38) with n = 2p, and (9.43) one gets that the integral formula
Z
(9.44)
R
√
2
x2p e−(x−x0 ) dx =
John Michael Rassias
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π
22p
2
· e−x0 ·
d2p x20
e ,
dx2p
0
holds for x0 ∈ R , and all p ∈ N0 .
If one substitutes √s2c , s ∈ R on x into the following general integral he finds
from (9.44) that
Z
Z
√
2
1
2p −2c(x−x0 )2
2p −(s− 2cx0 )
x e
dx =
·
s
e
ds
1
R
R
(2c)p+ 2
√
√
2
1
π −(√2cx0 )2
d2p
( 2cx0 )
=
·
·
e
·
e
√
1
2p
2p
d 2cx0
(2c)p+ 2 2
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√
=
(9.45)
π
1
1
2
· e−2cx0 ·
1
23p+ 2 cp+ 2
d2p
2cx2
√
2p e 0 , c > 0, x0 ∈ R,
d 2cx0
holds for all p ∈ N0 .
However,
d
d
dx0
1
d
√
(·) =
=
(·) √
(·) ,
1 ·
dx0
d 2cx0
d 2cx0
(2c) 2 dx0
and
d2p
1 d2p
(·) ,
√
2p (·) =
(2c)p dx2p
d 2cx0
0
(9.46)
On the Heisenberg-Pauli-Weyl
Inequality
c > 0, x0 ∈ R,
hold for all p ∈ N0 . Therefore from (9.45) and (9.46) we complete the proof of
(9.37).
Second, from Gasquet et al. [8, p.157-161] we claim that the Fourier transform fˆ : R → C is of the form
r
π − π2 ξ2 −i2πx0 ξ
(9.47)
fˆ (ξ) = c0
e c
, c0 ∈ C, c > 0, x0 ∈ R.
c
In fact, differentiating the Gaussian function f : R → C of the form f (x) =
2
c0 e−c(x−x0 ) with respect to x, one gets
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0
f (x) = −2c (x − x0 ) f (x) = −2cxf (x) + 2cx0 f (x) .
Page 127 of 151
0
Thus the Fourier transform of f is
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
∧
F f 0 (ξ) = F [f 0 (x)] (ξ) = [f 0 (x)] (ξ) = [−2cxf (x)]∧ (ξ) + [2cx0 f (x)]∧ (ξ) ,
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or
(9.48)
2iπξ fˆ (ξ) =
−2c
[(−2iπx) f (x)]∧ (ξ) + 2cx0 fˆ (ξ) ,
−2iπ
by standard formulas on differentiation, from Gasquet et al [8, p. 157]. Thus
c ˆ 0
f (ξ) + 2cx0 fˆ (ξ) ,
2iπξ fˆ (ξ) =
iπ
or
−2π 2 ξ fˆ (ξ) = cfˆ0 (ξ) + 2iπcx0 fˆ (ξ) ,
On the Heisenberg-Pauli-Weyl
Inequality
or
0
2π
(9.49)
fˆ (ξ) = fˆ0 (ξ) = − (πξ + icx0 ) fˆ (ξ) .
c
Solving the first order ordinary differential equation (9.49) by the method [25]
of the separation of variables we get the general solution
John Michael Rassias
(9.50)
2 2
ξ −i2πx
π
fˆ (ξ) = K (ξ) e− c
0ξ
,
such that fˆ (0) = K (0). Differentiating the formula (9.50) with respect to ξ
one finds
2
2π 2
0
− πc ξ 2 −i2πx0 ξ
0
ˆ
(9.51)
f (ξ) = e
ξ − i2πx0 .
K (ξ) + K (ξ) −
c
2
− πc ξ 2 −i2πx0 ξ
From (9.48), (9.49), (9.50) and (9.51) we find 0 = K 0 (ξ) e
K 0 (ξ) = 0, or
, or
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(9.52)
K (ξ) = K,
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which is a constant. But from (9.50) and (9.52) one gets
(9.53)
fˆ (0) = K (0) = K.
Besides from the definition of the Fourier transform we get
Z
ˆ
f (0) =
e−2iπ·0·x f (x) dx
ZR
=
f (x) dx
RZ
2
= c0 e−c(x−x0 ) dx
R
Z
√
2
√
c0
e−[ c(x−x0 )] d c (x − x0 ) ,
=√
c R
or
r
π
ˆ
(9.54)
f (0) = c0
, c0 ∈ C, c > 0.
c
From (9.53) and (9.54) one finds
r
π
(9.55)
K = c0
, c0 ∈ C, c > 0.
c
Therefore from (9.50) and (9.55) we complete the proof of the formula (9.47).
Another proof of (9.47) is by employing the formula (9.2) for the special Gaus2
sian φ (x) = c0 e−cx , such that
r
π − π2 ξ 2
φ̂ (ξ) = c0
e c , c0 ∈ C, c > 0.
c
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In fact, f (x) = φ (x − x0 ), or
fˆ (ξ) = [φ (x − x0 )]∧ (ξ)
Z
=
e−2iπξx φ (x − x0 ) dx
ZR
=
e−2iπξ(x+x0 ) φ (x) dx
R
(with x + x0 on x)
On the Heisenberg-Pauli-Weyl
Inequality
= e−2iπξx0
Z
e−2iπξx φ (x) dx = e−2iπξx0 φ̂ (ξ) ,
John Michael Rassias
R
or
fˆ (ξ) = e
−2iπx0 ξ
r
c0
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π − π2 ξ 2
e c ,
c
Contents
establishing (9.47).
Therefore
from (8.36) – (8.37) with xm = 0, from (9.41), and the mean of ξ
2
ˆ
for f of the form
Z
ξm =
2
π
ξ fˆ (ξ) dξ = |c0 |2
c
R
Z
ξ · e−2
π2 2
ξ
c
dξ = 0,
R
as well as from (9.1), (9.37) and (9.47), one finds that the left-hand side of the
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inequality (8.40) of Corollary 8.2 is
(m2p )|f |2 (m2p ) fˆ 2
| |
Z
Z
2 2
2p
2p ˆ
=
x |f (x)| dx ·
ξ f (ξ) dξ
R
R
Z
Z
2
π
π2
4
2p −2c(x−x0 )
2p −2c∗ ξ 2
∗
= |c0 |
ξ e
dξ
where c =
x e
dx ·
c
c
R
R
!
√
Γ p + 12
1
d2p 2cx20
π
π
−2cx20
·
·
e
·
e
= |c0 |4
1
c 24p+ 12 c2p+ 12
dx2p
(2c∗ )p+ 2
0
√
1
π
|c0 |4 −2cx20 d2p 2cx20
∗ 2p
Γ p+
· p+1 · e
e ,
= Hp
23p−1
2
c
dx2p
0
√ with Hp∗ = 1 2π p 2 holds for all p ∈ N, c0 ∈ C, c > 0, and x0 ∈ R.
Finally from the right-hand side of the inequality (8.40) of Corollary 8.2 with
[p/2]
Ep,f =
X
q=0
Z
εp,q
2
x f (q) dx
2q
R
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such that |Ep,f | < ∞ and
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(m2p )|f |2 (m2p ) fˆ 2 ≥ Hp∗
| |
2p
2
Ep,f
,
for any fixed but arbitrary p ∈ N, one completes the proof of the extremum
principle (9.36).
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Corollary 9.4. Assume that the Euler gamma function Γ is defined by (9.32).
2
Consider the Gaussian function f : R → C of the form f (x) = c0 e−c(x−x0 ) ,
where c0 , c,x0 are fixed but arbitrary constants and c0 ∈ C, c > 0, x0 ∈ R.
Assume that xm is the mean of x for |f |2 . Consider the Fourier transform
fˆ :
2
R → C of f , given as in the abstract, and ξm the mean of ξ for fˆ . Consider
2
the 2q th moment of x for f (q) by
Z
2
(µ2q ) f (q) 2 =
(x − xm )2q f (q) (x) dx,
| |
R
the constants εp,q as in (8.38), and
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
[p/2]
Ep,f =
X
q=0
εp,q (µ2q ) f (q) 2 ,
| |
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 , and any fixed but arbitrary p ∈ N. If
!
r
2
π
|c
|
√0
E|f∗ |2 = 1 − E|f |2 = 1 −
2 c
and
x∗0
=
x0 E|f∗ |2 ,
then the extremum principle
√
4
(9.56) |Ep,f | ≤
2
π
3p−1
2
·Γ
1
2
2p
21
2
d
1 |c0 |2 ∗ −p −cx∗2
∗
p+
·
· E|f |2
·e 0 ·
e2cx0
,
2 c p+1
2
dx2p
0
holds for any fixed but arbitrary p ∈ N.
Equality holds for p = 1; x0 = 0, or for p = 1; E|f |2 = 1.
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We note that xm = 0 only if x0 = 0, while in the previous Corollary 9.3
we have xm = 0 even if x0 6= 0.We may call E|f |2 and E|f∗ |2 complementary
probability (or energy) integrals.
Proof. It is clear, that from (9.1), (9.37), (9.39), (9.40) and (9.47) that
(µ2p )|f |2 (m2p ) fˆ 2
| |
Z 2p
2
=
x − x0 E|f |2
|f (x)| dx
R
Z
2 2p ˆ
2
×
ξ f (ξ) dξ
where xm = x0 E|f |
R
Z 2p
|c0 |4
−2c(x−x0 )2
=
π
x − x0 E|f |2
e
dx
c
R
Z
π2
2p −2c∗ ξ 2
∗
.
×
ξ e
dξ
where c =
c
R
∗
2
2
By placing x + x0 E|f | on x and letting x0 = x0 1 − E|f | = x0 E|f∗ |2 we have
On the Heisenberg-Pauli-Weyl
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(µ2p )|f |2 (m2p ) fˆ 2
| |
=π
=π
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4
Z
4
√
|c0 |
c
|c0 |
c
∗ 2
x2p e−2c(x−x0 ) dx
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(2c )
R
π
4p+ 12
2
Γ
!
p + 12
1
∗ p+ 2
1
c
2p+ 12
∗2
· e−2cx0 ·
2p
2
d
2cx∗0
e
2p
dx∗0
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!
Γ p + 12
1
(2π 2 /c)p+ 2
.
J. Ineq. Pure and Appl. Math. 5(1) Art. 4, 2004
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However,
d2p
1
d2p
(·)
=
2p
2p
2p (·) ,
dx∗0
dx
∗
0
E|f |2
and
Hp∗
2p
=
1
22(p+1) π 2p
.
Therefore
(9.57)
(m2p )|f |2 (m2p ) fˆ 2
| |
√
1
π
∗ 2p
= Hp
Γ p+
23p−1
2
4 −2p
d2p
|c0 |
∗2
∗2
× p+1 E|f∗ |2
· e−2cx0 · 2p e2cx0
c
dx0
2p
≥ Hp∗ |Ep,f |2 from our above theorem ,
completing the proof of Corollary 9.4.
9.3.
Two Special Cases of (9.56)
(i) If p = 1, then
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dx∗0
∗
4cx0 e
dx0
∗ 2
2
d
dx0
∗ 2cx∗0
= 4c ∗ x0 e
dx0
dx0
d2 2cx∗02
d
e
=
2
dx0
dx0
2
2cx∗0
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=
2
4cE|f∗ |2
1+
2
4cx∗0
∗2
e2cx0 , where
∗2
dx∗0
= E|f∗ |2 .
dx0
2
Therefore at x0 = 0 (or x∗0 = 0), d2 e2cx0 /dx20 = 4cE|f∗ |2 . If we denote by
R.H.S. the right hand side of (9.56), then
(9.58)
R.H.S. (for p = 1; x0 = 0 or x∗0 = 0)
s √
4
π
3 |c0 |2 ∗ −1 q
2
Γ
E|f |2
4cE|f∗ |2
=
2
2
c
r
2
π |c0 |
√ = E|f |2 .
=
2 c
We note at x = x0 = 0 one can get from (9.39) that xm = x0 E|f |2 = 0.
But we have ξm = 0. Therefore from (8.26) one finds that
(9.59)
|E1,f | = E|f |2 .
Thus from (9.58) and (9.59) we establish the equality in (9.56) for p = 1
and x0 = 0.This corresponds to the equality of (9.36), as well.
Besides we note from (9.56) at x0 6= 0 one gets that
(9.60)
R.H.S. (for p = 1; x0 6= 0)
s √
i 12
4
3 |c0 |2 ∗ −1 h
π
∗2
∗2
E|f |2
4cE|f |2 1 + 4cx0
=
Γ
2
c
2
2 12
= E|f |2 1 + 4c 1 − E|f |2 x20 .
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Inequality
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In this case from (9.39) we have that xm = x0 E|f |2 6= 0. But ξm = 0.
Therefore from (8.51) one finds for p = 1; q = 0, and w = 1 that
Z
E1,f = − |f (x)|2 dx = −E|f |2
R
satisfying (8.26).
Thus from (9.59) and (9.60) one establishes the inequality in (9.56) for
p = 1 and both x0 6= 0 and E|f |2 6= 1, such that
(9.61)
|E1,f | = E|f |2
2 21
≤ E|f |2 1 + 4c 1 − E|f |2 x20 .
If either x0 = 0, or E|f |2 = 1, then the equality in (9.56) holds for p = 1.
2 2
d4 2cx∗02
∗2 d
∗2
2cx∗0
e
=
4cE
1
+
4cx
e
2
0
|f | dx2
dx40
0
d ∗
3
3
∗2
= 16c2 E|f∗ |2
3x0 + 4cx∗0 e2cx0
dx0
4
2
4
∗2
= 16c2 E|f∗ |2 3 + 24cx∗0 + 16c2 x∗0 e2cx0 .
Therefore at x0 = 0 (or x∗0 = 0), we have
(9.63)
John Michael Rassias
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(ii) If p = 2, then one gets
(9.62)
On the Heisenberg-Pauli-Weyl
Inequality
d4 2cx∗02
4
e
= 48c2 E|f∗ |2 .
4
dx0
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Thus
R.H.S. (for p = 2; x0 = 0 (or x∗0 = 0))
s √
4
π
5 |c0 |2 ∗ −2 q 2 ∗4
= 5 Γ
E|f |2
48c E|f |2 ,
2 c 32
22
or
(9.64)
3
R.H.S. =
2
r
3
π |c0 |2
√ = E|f |2 , for p = 2; x0 = 0.
2
2 c
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
We note at x = x0 = 0 one can get from (9.39) that xm = 0. But we have
ξm = 0. Therefore from (8.27) one finds that
Z h
i
3
2
2
2
0
E2,f = 2
|f (x)| − x |f (x)| dx = 2 E|f |2 − E|f |2 ,
4
R
or
(9.65)
1
|E2,f | = E|f |2 .
2
Thus from (9.64) and (9.65) we establish the inequality in (9.56) for p = 2
and x0 = 0, because
(9.66)
1
3
|E2,f | = E|f |2 < E|f |2 .
2
2
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Finally we note from (9.56) at x0 6= 0 one gets that
(9.67) R.H.S. (for p = 2; x0 6= 0)
2
4 12
3
= E|f |2 3 + 24c 1 − E|f |2 x20 + 16c2 1 − E|f |2 x40 .
2
We note that
Z
2
(x − xm )2 |f 0 (x)| dx
R
Z 2
2
= 4c
x − x0 E|f |2 (x − x0 )2 |f (x)|2 dx
R
Z
Z
2
2
4
= 4c
x |f (x)| dx − 2 1 + E|f |2 x0 x3 |f (x)|2 dx
R
R
Z
2
+ 1 + 4E|f |2 + E|f2 |2 x20 x2 |f (x)| dx
R
Z
Z
2
2
2
2
4
3
− 2 E|f |2 + E|f |2 x0 x |f (x)| dx +E|f |2 x0 |f (x)| dx .
R
R
But (9.37) holds even if we replace 2p with any fixed but arbitrary n ∈ N0
(from (9.38)). Then one gets that,
Z
|f (x)|2 dx = E|f |2 ,
Z R
x |f (x)|2 dx = xm = x0 E|f |2 ,
R
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Z
ZR
x3 |f (x)|2 dx = (m3 )|f |2
R
and
Z
1 + 4cx20
E|f |2 ,
4c
3x0 + 4cx30
=
E|f |2 ,
4c
x2 |f (x)|2 dx = (m2 )|f |2 =
x4 |f (x)|2 dx = (m4 )|f |2 =
R
3 + 24cx20 + 16c2 x40
E|f |2
16c2
2
hold, if f (x) = c0 e−c(x−x0 ) , c0 ∈ C, c > 0, x0 ∈ R. Therefore
Z
2
(x − xm )2 |f 0 (x)| dx
R
1
= E|f |2 3 + 24cx20 + 16c2 x40 − 8c 1 + E|f |2 3x20 + 4cx40
4
+ 4c 1 + 4E|f |2 + E|f2 |2 x20 + 4cx40
i
−32c2 E|f |2 + E|f2 |2 x40 +16c2 E|f2 |2 x40
n
h
i
1
2
= E|f |2 3 + 4c 6 − 6 1 + E|f |2 + 1 + 4E|f |2 + E|f |2 x20
4
h
+ 162 1 − 2 1 + E|f |2 + 1 + 4E|f |2 + E|f2 |2
i o
2
2
− 2 E|f |2 + E|f |2 + E|f |2 x40 ,
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John Michael Rassias
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or
Z
2
0
2
(x − xm ) |f (x)| dx =
R
3 + 4c 1 − E|f |2
4
2
x20
E|f |2 ,
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holds, if c > 0, x0 ∈ R.
In this case from (9.39) we have that xm = x0 E|f |2 6= 0. But ξm = 0.
Therefore from (8.51) one finds for p = 2; q = 0, 1 and w = 1 that
Z
2
2! 2
2−0
2−0
E2,f = (−1)
(1)(0)
0
0
2−0
R 0!
2
0
× (x − xm ) (x − xm )2·0 f (0) (x) dx
Z
2! 0
2
2−1
2−1
+ (−1)
(1)(0)
1
0
2−1
2!
R
2
0
2·1 (1)
× (x − xm ) (x − xm ) f (x) dx
Z h
i
2
=2
|f (x)|2 dx − (x − xm )2 |f 0 (x)| dx
R

2
2
3 + 4c 1 − E|f |2 x0


= 2 E|f |2 −
E|f |2  ,
4
or
(9.68)
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
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E2,f
2 1
= E|f |2 1 − 4c 1 − E|f |2 x20 .
2
Thus from (9.67) and (9.68) one establishes the inequality in (9.56) for
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p = 2 such that
(9.69)
2 1
|E2,f | = E|f |2 1 − 4c 1 − E|f |2 x20 2
2
3
< E|f |2 3 + 24c 1 − E|f |2 x20
2
4 12
2
+16c 1 − E|f |2 x40 ,
because the condition
2 2 2
2
4 4c 1 − E|f |2 x0 + 28 4c 1 − E|f |2 x20 + 13 > 0,
On the Heisenberg-Pauli-Weyl
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John Michael Rassias
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or
(9.70)
Contents
64c2 1 − E|f |2
4
x40 + 112c 1 − E|f |2
2
x20 + 13 > 0
holds for p = 2 and for fixed but arbitrary constants c > 0, and x0 ∈ R.
If either x0 = 0, or E|f |2 = 1, then we still have inequality in (9.56) for
p = 2.
Corollary 9.5. Assume that the Gamma function Γ is defined by (9.32). Con2
sider the general Gaussian function f : R → C of the form f (x) = c0 e−c1 x +c2 x+c3 ,
where ci , (i = 0, 1, 2, 3) are fixed but arbitrary constants and c0 ∈ C, c1 >
0,
c2
and c2 , c3 ∈ R. Assume that the mathematical expectation E x − 2c1 of
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x−
c2
2c1
for |f |2 equals to
Z xm =
R
c2
x−
2c1
|f (x)|2 dx = 0.
Consider the Fourier transform
fˆ : R → C of f , given as in the abstract, and
2
2
ξm the mean of ξ for fˆ . Denote by (m2q ) f (q) 2 the 2q th moment of x for f (q) | |
about the origin, as in (8.41), and the constants εp,q as in (8.38). Consider
On the Heisenberg-Pauli-Weyl
Inequality
[p/2]
Ep,f =
X
q=0
2
If ε0 = c0 e(c2 +4c1 c3 )/4c1 ∈ C and t0 =
√
4
(9.71)
|Ep,f | ≤
2
π
3p−1
2
·Γ
1
2
p ∈ N.
εp,q (m2q ) f (q) 2 ,
| |
c2
2c1
1
p+
2
John Michael Rassias
∈ R, then the extremum principle
·
|ε0 |2
p+1
−c1 t20
·e
c1 2
·
d2p 2c1 t20
e
dt2p
0
Contents
12
,
holds for any fixed but arbitrary p ∈ N.
Equality holds for p = 1; t0 = 0 (or c2 = 0).
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We note that xm = 0 even if t0 6= 0 or c2 6= 0, while in the following
Corollary 9.6 we have xm = 0 only if t0 = 0 or c2 = 0.
2
2
Also we observe that if g (x) = c0 e−c1 (x−t0 ) , and d0 = e(c2 +4c1 c3 )/4c1 (> 0),
then
(9.72)
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2
f (x) = d0 g (x) = ε0 e−c1 (x−t0 ) ,
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ε0 ∈ C.
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Proof. In fact, from (9.36) and (9.72) one gets that
c2
xm = E x −
2c1
Z c2
x−
=
|f (x)|2 dx
2c1
R
Z c2
2
= d0
x−
|g (x)|2 dx
2c1
R
Z
Z
c2
2
2
2
= d0
x |g (x)| dx −
|g (x)| dx
2c1 R
R
c2
c2
2
= d0
E 2−
E 2 ,
2c1 |g|
2c1 |g|
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
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or
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(9.73)
xm = 0.
2
In this case the mathematical expectation xm is the mean of x for |f | if t0 = 0
or c2 = 0.
We note that from (9.40) and (9.72) one establishes
r
r
1 c3
π |c0 |2 c22 +4c
π |c0 |2
2
2
(9.74)
E|f |2 = d0 E|g|2 = d0
√ e 2c1 .
√ =
2 c1
2 c1
This result can be computed directly from (9.40), as follows:
r
π |ε0 |2
E|f |2 =
√
2 c1
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which leads to (9.74).
Similarly from (9.39) we get the mean of f for |f |2 of the form
Z
(9.75)
x |f (x)|2 dx = t0 E|f |2
R
r
π |ε0 |2
=
√ t0
2 c1
r
1 π |c0 |2 c2 (c22 +4c1 c3 )/2c1
.
=
√ e
2 2 c1 c1
Also from (9.47) one finds the Fourier transform fˆ of f of the form
r
π − πc 2 ξ2 −i2πt0 ξ
ˆ
(9.76)
f (ξ) = ε0
e 1
c1
r
π (c22 +4c1 c3 )/4c1 −(π2 ξ2 +iπc2 ξ)/c1
e
,
= c0
e
c1
c0 ∈ C, c1 > 0, and c2 , c3 ∈ R.
2
Finally we find the mean of ξ for fˆ , as follows:
Z
(9.77)
Z
2
ˆ 2
−2 π ξ 2
2 (c22 +4c1 c3 )/2c1 π
ξm =
ξ f (ξ) dξ = |c0 | e
ξ · e c1 dξ = 0.
c1 R
R
The rest of the proof is similar to the proof of the Corollary 9.3.
Corollary 9.6. Assume the gamma function Γ,given as in (9.32). Consider the
2
general Gaussian function f : R → C of the form f (x) = c0 e−c1 x +c2 x+c3 ,
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John Michael Rassias
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where ci , (i = 0, 1, 2, 3) are fixed but arbitrary constants and c0 ∈ C, c1 > 0,
and c2 , c3 ∈ R. Assume that xm is the mean (or the mathematical expectation
E (x)) of x for |f |2 . Consider the Fourier transform
fˆ : R → C of f , given as
2
in the abstract, and ξm the mean of ξ for fˆ .
2
Consider (µ2q ) f (q) 2 the 2q th moment of x for f (q) by
| |
Z
2
(µ2q ) f (q) 2 =
(x − xm )2q f (q) (x) dx,
| |
R
the constants εp,q as in (8.38), and
On the Heisenberg-Pauli-Weyl
Inequality
John Michael Rassias
[p/2]
Ep,f =
X
q=0
εp,q (µ2q ) f (q) 2 ,
| |
if |Ep,f | < ∞
Title Page
Contents
holds for 0 ≤ q ≤ p2 , and any fixed but arbitrary p ∈ N. If
2
ε0 = c0 e(c2 +4c1 c3 )/4c1 ∈ C,
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c2
t0 =
∈ R,
2c1
E|f∗ |2
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r
= 1 − E|f |
2
=1−
π |ε0 |2
√
2 c1
!
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and t∗0 = t0 E|f∗ |2 , then the following extremum principle
√
4
(9.78) |Ep,f | ≤
2
π
3p−1
2
·Γ
1
2
12
1 |ε0 |2 ∗ −p −c1 t∗0 2 d2p 2c1 t∗2
p+
·
· E|f |2
·e
·
e 0
,
2 c p+1
2
dt2p
0
1
holds for any fixed but arbitrary p ∈ N.
Equality holds for p = 1; t0 = 0 (or c2 = 0), or for p = 1; E|f |2 = 1.
We note that xm = 0 only if t0 = 0, while in the previous Corollary 9.5 we
have xm = 0 even if t0 6= 0.
p π |ε0 |2
√ .
From (9.74) – (9.75) one gets that xm = t0 E|f |2 , where E|f |2 =
2 c1
Thus we get from (9.76) – (9.77) the proof of Corollary 9.6, in a way similar to the proof of the Corollary 9.4, because from (9.72) we have f (x) =
2
ε0 e−c1 (x−t0 ) , and xm is the mean of x for |f |2 .
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References
[1] G. BATTLE, Heisenberg Inequalities for Wavelet States, Appl. Comp.
Harm. Analysis, 4 (1997), 119.
[2] R.N. BRACEWELL, The Fourier transform and its Applications,
McGraw-Hill, New York, 1986.
[3] B. BURKE HUBBARD, The World according to Wavelets, the Story of a
mathematical technique in the making, AKPeters, Natick, Massachusetts,
1998.
[4] R.R. COIFMAN, Y. MEYER, S. QUAKE AND M.Y. WICKERHAUSER,
Signal Processing and Compression with Wavelet Packets, in Progress in
Wavelet Analysis and Applications, Y. Meyer and S. Roques, eds., Proc. of
the Intern. Conf. on Wavelets and Applications, Toulouse, France, Editions
Frontières, 1993.
[5] I. DAUBECHIES, Ten Lectures on Wavelets, Vol. 61, SIAM, Philadelphia,
1992.
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John Michael Rassias
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[6] G.B. FOLLAND AND A. SITARAM, The Uncertainty Principle: A Mathematical Survey, J. Fourier Anal. & Appl., 3 (1997), 207.
Go Back
[7] D. GABOR, Theory of Communication, Jour. Inst. Elect. Eng., 93 (1946),
429.
Quit
[8] C. GASQUET AND P. WITOMSKI, Fourier Analysis and Applications,
(Springer-Verlag, New York, 1998).
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[9] W. HEISENBERG, Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik, Zeit. Physik, 43 (1927), 172; The Physical
Principles of the Quantum Theory, Dover, New York, 1949; The Univ.
Chicago Press, 1930.
[10] A. KEMPF, Black holes, bandwidths and Beethoven, J. Math. Phys., 41
(2000), 2360.
[11] J. F. MULLIGAN, Introductory College Physics, McGraw-Hill Book Co.,
New York, 1985.
[12] L. RAYLEIGH, On the Character of the Complete Radiation at a given
temperature, Phil. Mag., 27 (1889); Scientific Papers, Cambridge University Press, Cambridge, England, 1902, and Dover, New York, 3, 273
(1964).
[13] F.M. REZA, An Introduction to Information Theory, Dover, New York,
1994 and McGraw-Hill, New York, 1961.
[14] N. SHIMENO, A note on the Uncertainty Principle for the Dunkl Transform, J. Math. Sci. Univ. Tokyo, 8 (2001), 33.
[15] R.S. STRICHARTZ, Construction of orthonormal wavelets, Wavelets:
Mathematics and Applications, (1994), 23–50.
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John Michael Rassias
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[16] E.C. TITCHMARSH, A contribution to the theory of Fourier transforms,
Proc. Lond. Math. Soc., 23 (1924), 279.
[17] G.P. TOLSTOV, Fourier Series, translated by R.A. Silverman, PrenticeHall, Inc., Englewood Cliffs, N.J., 1962.
Page 148 of 151
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[18] F.G. TRICOMI, Vorlesungen über Orthogonalreihen, Springer-Verlag,
Berlin, 1955.
[19] G.N. WATSON, A Treatise on the Theory of Bessel Functions, Cambridge
University Press, London, 1962.
[20] H. WEYL, Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig,
1928; and Dover edition, New York, 1950.
[21] E.T. WHITTAKER AND G.N. WATSON, A Course of Modern Analysis,
Cambridge University Press, London, 1963.
[22] N. WIENER, The Fourier integral and certain of its applications, Cambridge, 1933.
[23] N. WIENER, I am a Mathematician, MIT Press, Cambridge, 1956.
[24] J.A. WOLF, The uncertainty principle for Gelfand pairs, Nova J. Algebra
Geom., 1 (1992), 383.
[25] D.G. ZILL, A First Course in Differential Equations with Applications,
Prindle, Weber & Schmidt, Boston, Mass., 1982.
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Table 1: The first thirty-three cases of (9.33) and R∗ (p) = 2πR (p) ≥ 1
p
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
R(p)
0.16
0.48
0.80
0.43
0.59
1.93
2.97
1.68
2.34
7.80
11.63
6.65
9.33
31.30
46.04
26.52
37.26
125.48
183.10
105.83
148.89
R*(p)
1.00
3.00
5.00
2.69
3.71
12.16
18.65
10.53
14.70
48.98
73.06
41.81
58.61
196.66
289.30
166.61
234.09
788.41
1150.43
664.95
935.48
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p
22
23
24
25
26
27
28
29
30
31
32
33
R(p)
502.68
729.57
422.69
595.18
2012.88
2910.37
1688.95
2379.65
8058.08
11617.84
6750.36
9515.51
R*(p)
3158.42
4584.05
2655.83
3739.60
12647.30
18286.41
10611.98
14951.80
50630.40
72997.06
42413.73
59787.71
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