Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 1, Article 4, 2004
ON THE HEISENBERG-PAULI-WEYL INEQUALITY
JOHN MICHAEL RASSIAS
P EDAGOGICAL D EPARTMENT, E. E.,
NATIONAL AND C APODISTRIAN U NIVERSITY OF ATHENS ,
S ECTION OF M ATHEMATICS AND I NFORMATICS ,
4, AGAMEMNONOS S TR ., AGHIA PARASKEVI ,
ATHENS 15342, G REECE .
jrassias@primedu.uoa.gr
Received 02 January, 2003; accepted 04 March, 2003
Communicated by A. Babenko
A BSTRACT. 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
2
2
moment of the random real x for |f | and the second moment of the random real ξ for fˆ is
R
at least E|f |2 /4π , where fˆ is the Fourier transform of f , such that fˆ (ξ) = R e−2iπξx f (x) dx
R
R
√
2
and f (x) = R e2iπξx fˆ (ξ) dξ, i = −1 and E|f |2 = R |f (x)| dx. In this paper we generalize the afore-mentioned result to the higher moments for L2 functions f and establish the
Heisenberg-Pauli-Weyl inequality.
Key words and phrases: Pascal Identity, Plancherel-Parseval-Rayleigh Identity, Lagrange Identity, Gaussian function, Fourier
transform, Moment, Bessel equation, Hermite polynomials, Heisenberg-Pauli-Weyl Inequality.
2000 Mathematics Subject Classification. Primary: 26Xxx; Secondary: 42Xxx, 60Xxx, 62Xxx.
1. I NTRODUCTION
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:
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
We thank the computer scientist C. Markata for her help toward the computation of the data of Table 9.1 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.
025-03
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J OHN M ICHAEL R ASSIAS
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
2following proposition holds: the less the
2
uncertainty in |f | , the greater the uncertainty in fˆ 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) .
1.1. Second Order Moment Heisenberg Inequality ([3, 6]). For any f ∈ L2 (R), f : R → C,
such that
Z
kf k22 =
|f (x)|2 dx = E|f |2 ,
R
any fixed but arbitrary constants xm , ξm ∈ R, and for the second order moments (variances)
Z
2
(µ2 )|f |2 = σ|f |2 =
(x − xm )2 |f (x)|2 dx
R
and
(µ2 ) fˆ =
| |
2
σ2ˆ 2
|f |
Z
ˆ 2
(ξ − ξm ) f (ξ) dξ,
2
=
R
the second order moment Heisenberg inequality
(H1 )
E|f2 |2
(µ2 )|f |2 · (µ2 ) fˆ 2 ≥
,
| |
16π 2
holds, where
fˆ (ξ) =
Z
e−2iπξx f (x) dx
R
and
Z
f (x) =
e2iπξx fˆ (ξ) dξ, i =
√
−1.
R
Equality holds in (H1 ) iff (if and only if) the Gaussians
2
f (x) = c0 e2πixξm e−c(x−xm ) = c0 e−cx
2 +2(cx
2
m +iπξm )x−cxm
hold for some constants c0 ∈ C and c > 0. We note that if xm 6= 0 and ξm = 0, then
2
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. Besides
2
d
0
¯
we consider both the ordinary derivative dx |f | = 2 Re f f 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
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
3
implies f 0 ∈ L2 . Integration by parts yields
a+
Z a+
Z a+
d
2
2
0
2
x Re f (x) f (x) dx =
x |f (x)| dx = x |f (x)| dx
a−
a−
a−
Z a+
−
|f (x)|2 dx, −∞ < a− < a+ < ∞.
a−
0
2
Since f , xf , f ∈ L , 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
2
(m2 )|f |2 = s|f |2 =
x2 |f (x)|2 dx
R
and
(m2 ) fˆ 2 =
| |
s2 ˆ 2
|f |
Z
=
R
ˆ 2
ξ f (ξ) dξ,
2
one gets
1
kf k42 =
2
16π
=
=
=
=
≤
≤
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
0
2 x Re f (x) f (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) of first order
2
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 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 )
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
2
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J OHN M ICHAEL R ASSIAS
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 =
dfa
dfa dxδ
dfa
=
=
,
dx
dxδ dx
dxδ
xδ = x − x m .
In fact,
ln |fa | = −c (x − xm )2 ,
or
2
eax f (x) = fa (x) = c0 e−c(x−xm ) ,
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
We note that, if f ∈ L2 (R) and the L2 -norm of f is kf k2 = 1 = fˆ , then |f |2 and fˆ
2
are both probability density functions. The Heisenberg inequality in mathematical statistics
and Fourier analysis
asserts that: The product of the variances of the probability measures
ˆ 2
2
|f (x)| dx and f (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
2
of the 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:
1
2
σ|f
· σ 2 ˆ 2 ≥ kf k42 ,
|2
|f |
4
if f ∈ L2 (R),
Z
1
ˆ
f (ξ) = √
e−iξx f (x) dx
2π R
and
Z
1
f (x) = √
eiξx fˆ (ξ) dξ,
2π R
2
where the L -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 bandwidth ∆ξ of a.signal cannot
2
be less than the value 41 π = H ∗ (=Heisenberg lower bound), where ∆x2 = σ|f
E|f |2 and
|2
.
2
2
2
∆ξ = σ ˆ 2 E fˆ 2 = σ ˆ 2 E|f |2
| |
|f |
|f |
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.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
http://jipam.vu.edu.au/
O N THE H EISENBERG -PAULI -W EYL INEQUALITY
5
2. PASCAL T YPE C OMBINATORIAL I DENTITY
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 i ∈ N0 =
k
{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−1 k−i
k
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
completing the proof of this identity.
Note that all of the three combinations:
numbers if 1 ≤ i ≤ k2 for i ∈ N.
k−i
i
,
k−i
i−1
, and
k−i+1
i
exist and are positive
3. G ENERALIZED D IFFERENTIAL I DENTITY
We state and prove the new differential identity.
Proposition 3.1. If f : R → C is a complex valued function of a real variable x, 0 ≤
dj
greatest integer ≤ k2 , f (j) = dx
j f , and (·) is the conjugate of (·), then
(*)
f (x) f
(k)
(x) + f
(k)
(x) f¯ (x) =
[ k2 ]
X
i=0
k
(−1)
k−i
i
k−i
i
2
is the
2
dk−2i (i)
,
f
(x)
dxk−2i
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤
{0, 1, 2, . . .}.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
k
k
2
for i ∈ N0 =
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6
J OHN M ICHAEL R ASSIAS
Note that for k = 1 we have
f (x) f
(1)
(x) + f
(1)
1
(x) f¯ (x) = (−1)[ 2 ](=0)
1
1−0
1−0
0
2
d1−2·0 (0)
f
(x)
dx1−2·0
d
|f (x)|2
dx
(1)
= |f (x)|2
.
=
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
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)
(1) (k)
(k+1)
(k+1) ¯
(k) (1)
= f f + ff
+ f
f +f f
= f (1) f (k) + f (k) f (1) + f f (k+1) + f (k+1) f¯
(k−1)
(1)
(1) (k−1) (1)
(1)
= f (f )
+ f
f
+ Gk+1 (f )
= Gk−1 f (1) + Gk+1 (f ) ,
or the recursive sequence
(1)
Gk+1 (f ) = Gk (f ) − Gk−1 f (1) ,
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
(R)
j
X
λi+1 =
i=0
j+1
X
λi , (−1)i−1 = − (−1)i
i=1
for i ∈ N0 , and
 k
 2 − 1,
k−1
=
 k
2
2
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
=
k−1
2
k = 2v for v = 1, 2, . . .
, k = 2λ + 1 for λ = 0, 1, . . .
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
such that
k
2
≤
k−1 2
7
+ 1, if k ∈ N, we find
Gk+1 (f )


k

[
]
2

X
2 (k+1−2i) 
k
k−i
i
f (i) =
(−1)
i


k−i

 i=0


k−1


[
]
2
X
(k−1−2i) 
k−1
k−1−i
i
(i+1) 2
−
(−1)
f
i


k−1−i

 i=0
(0) 2 (k+1−2·0)
k
k−0
0
f = (−1)
0
k−0

k

[X
2]
(i) 2 (k+1−2i) 
k
k−i
i
(−1)
f
+
i

k
−
i

i=1


k−1
+1


2 ]
[ X
(i) 2 (k−1−2(i−1)) 
k−1
k − 1 − (i − 1)
i−1
f −
(−1)
i
−
1


(k
−
1)
−
(i
−
1)
 i=1





[ k2 ]

X
(i) 2 (k+1−2i) 
k
(k+1)
k−i
2
i
f =
|f |
+
(−1)
i


k
−
i


i=1


k−1
+1

[
]
2
 X

2 (k+1−2i) 
k−i
i k−1
f (i) +
(−1)
i−1


k−i
 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




[ k2 ]
X

(i) 2 (k+1−2i)
k−i
i k−1
f +
(−1)
+ S (f ) ,
i−1


k−i
 i=1

where

0,
if k = 2v for v = 1, 2, . . .







k−1 
+1
[ k−1

2 ]
(−1)
(k
−
1)

+
1
k
−

k−1 2 +1 −1
+
1
k − k−1
S (f ) =
2
2


+1))
2 (k+1−2([ k−1

2 ]

k−1

+1
]
)
([

2
,
×
f





if k = 2λ + 1 for λ = 0, 1, . . . .
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J OHN M ICHAEL R ASSIAS
Thus
[ k2 ]
X
k
k−i
2 (k+1)
i
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 ) =
=
=

0,





if k = 2v for v = 1, 2, . . .
k−1

(−1)[ 2 ]




1−k
k−1−[ k−1
2 ]

0,



2 (k−1−2[ k−1
2 ])
k − 1 − k−1
([ k−1
+1) ]
2
2
f
,
k−1
2
if k = 2λ + 1 for λ = 0, 1, . . .
if k = 2v for v = 1, 2, . . .
k−1


 (−1) 2

0,



1−k
k−1
2
k−1
2
k−1
2
2 (0)
( k+1
)
, if k = 2λ + 1 for λ = 0, 1, . . .
f 2 if k = 2v for v = 1, 2, . . .
2 k+1
( k+1

)
2

2
f
, if k = 2λ + 1 for λ = 0, 1, . . .
 2 (−1)

0,if k = 2v for v = 1, 2, . . .





k+1 2 (k+1−2[ k+1
2 ])
=
k+1
k+1
k
+
1
−
[
]
k+1
([
])
2
2
2

(−1)
f
,

k+1

k+1−[ k+1
2 ]

2

if k = 2λ + 1 for λ = 0, 1, . . . ,
because
k+1
k+1
=
, if k = 2λ + 1 for λ = 0, 1, . . .
2
2
Besides
k+1 we note that, from the above Pascal type combinatorial identity (C) and
, if k = 2v for v = 1, 2, . . . , one gets
2
k
2
=
k
2
=
k ) k 2 (k+1−2[ k2 ])
k
k
−
1
k
−
k
−
k 2
k k 2
+
(−1)[ ]
f ([ 2 ]) −1
k− 2
2
2
2 (k+1−2[ k2 ])
k
k
k+1
k − k2 + 1
[
]
])
([
2
k
= (−1)
f 2 k
k− 2 +1
2
2 (k+1−2[ k+1
2 ])
k+1
k+1
k+1
k + 1 − k+1
]
[
])
([
2
2
2
k+1 = (−1)
,
f
k+1
k+1− 2
2
(
k
2
k
k − k2
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
9
if k = 2v for v = 1, 2, . . . . From these results, one obtains
(0) 2 (k+1−2·0)
k+1
k+1−0
0
f Gk+1 (f ) = (−1)
0
k+1−2·0
[ k2 ]
X
(i) 2 (k+1−2i)
k+1
k+1−i
i
f +S (f )
+
(−1)
i
k+1−i
i=1
k+1
2
=
[X]
i=1
k+1
(−1)
k+1−i
i
k+1−i
i
(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.
3.1. Special cases of (**).
G1 (f ) = |f |2
G2 (f ) = |f |2
G3 (f ) = |f |2
G4 (f ) = |f |2
(4)
2 (5)
G5 (f ) = |f |
2 (6)
G6 (f ) = |f |
G7 (f ) = |f |2
G8 (f ) = |f |2
(8)
(7)
(2)
(3)
(1)
,
2
− 2 f (1) ,
2 (1)
− 3 f (1) ,
2
2 (2)
− 4 f (1) + 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) ,
2 (3)
2 (1)
2 (5)
+ 14 f (2) − 7 f (3) ,
− 7 f (1) 2 (6)
2 (4)
2 (2)
2
− 8 f (1) + 20 f (2) − 16 f (3) + 2 f (4) .
We note that if one takes the above numerical coefficients of Gi (f ) (i = 1, 2, . . ., 8) absolutely,
then one establishes the pattern
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
a1 = 2
a2 = 5
a3 = 9 b1 = 2
a4 = 14 b2 = 7
a5 = 20 b3 = 16 c1 = 2
with
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J OHN M ICHAEL R ASSIAS
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
c 1 = b 1 = a1 = 2
b 2 = a1 + a2 = 7
b3 = a1 + a2 + a3 = 16
Following this pattern we get
2 (7)
2 (5)
2 (3)
2 (1)
(9)
G9 (f ) = |f |2
− 9 f (1) + a6 f (2) − b4 f (3) + c2 f (4) ,
where
a6 = a5 + 7 = 27,
b4 = b3 + a4 = 16 + 14 = 30,
c2 = c1 + b2 = 2 + 7 = 9.
Similarly, one gets
2 (10)
G10 (f ) = |f |
(8)
(6)
(1) 2
(2) 2
− 10 f
+ a7 f
2
(4)
(2)
(3) 2
(4) 2
− b5 f
+ c3 f
− d1 f (5) ,
where
a7
b5
c3
d1
= a6 + 8 = 35,
= b4 + a5 = 30 + 20 = 50,
= c2 + b3 = 9 + 16 = 25,
= c1 = b1 = a1 = 2.
3.2. Applications of the Recursive Sequence (R).
(1)
G4 (f ) = G3 (f ) − G2 f (1)
(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
= |f |
−3 f
−
f
−2 f
= |f |2
(4)
2 (2)
2
− 4 f (1) + 2 f (2) and
(1)
G5 (f ) = G4 (f ) − G3 f (1)
(3)
(1)
(1) 2 (3)
(1)
2 (5)
(1) 2
(2) 2
(2) 2
= |f |
−4 f
+2 f
−
f
−3 f
= |f |2
(5)
2 (3)
2 (1)
,
− 5 f (1) + 5 f (2) yielding also the above generalized differential identity (**) for k = 3 and k = 4, respectively.
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
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3.3. Generalization of the Identity (**). We denote
Hkl (f ) = f (l) f (k) + f (k) f (l) .
It is clear that

(l)
(l)
(l) )(k−l) + f (l) (k−l) f (l) ,

G
f
=
f
(f
if k > l

k−l




2
Hkl (f ) =
G0 f (l) = 2 f (l) ,
if k = l





(l−k)

(l−k)
Gl−k f (k) = f (k) (f (k) )
+ f (k)
f (k) , if k < l
From these and (**) we conclude that

k−l
[X

2 ]

(i+l) 2 (k−l−2i)

k
−
l
k
−
l
−
i

i
f

(−1)
, 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
For instance, if k = 3 > 2 = l, then from this formula one gets
H32 (f ) = G3−2 f (2)
3−2
[X
2 ]
(i+2) 2 (3−2−2i)
3−2
3−2−i
f
=
(−1)
i
3
−
2
−
i
i=0
(2) 2 (1−2·0)
1
1−0
0
f = (−1)
0
1−0
(1)
(2) 2
= f
.
i
In fact,
H32 (f ) =
f (2) f (3)
+
f (3) f (2)
=f
(2)
(1)
(f (2) )
+ f
(2) (1)
f (2)
= G1 f
(2)
(1)
(2) 2
= f
.
Another special case, if k = 3 > 1 = l, then one gets
H31 (f ) = G3−1 f (1)
3−1
[X
2 ]
(i+1) 2 (3−1−2i)
3−1
3−1−i
f
=
(−1)
i
3
−
1
−
i
i=0
(1) 2 (2−2·0)
(2) 2 (2−2·1)
2
2
2−0
2
−
1
0
1
f = (−1)
f
+ (−1)
0
1
2−0
2−1
(2)
2
(1) 2
= f
− 2 f (2) .
i
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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12
J OHN M ICHAEL R ASSIAS
In fact,
H31 (f ) = f (1) f (3) + f (3) f (1)
(2)
(2)
= f (1) (f (1) ) + f (1)
f (1)
2
(1) 2 (2)
(1)
= G2 f
= f − 2 f (2) .
4. G ENERALIZED P LANCHEREL -PARSEVAL -R AYLEIGH I DENTITY
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 variable x,
√
(p)
dp
−1 and any fixed but arbitrary
fa = eax f and fa = dx
p fa , where a = −2πξm i with i =
ˆ
constant ξm ∈ R, f the Fourier transform of f , such that
Z
fˆ (ξ) =
e−2iπξx f (x) dx
R
with ξ real and
Z
f (x) =
e2iπξx fˆ (ξ) dξ,
R
as well as f ,
(p)
fa ,
(4.1)
and (ξ − ξm ) fˆ are in L2 (R), then
Z
Z
2
(p)
1
2p ˆ
fa (x)2 dx
(ξ − ξm ) f (ξ) dξ =
2p
(2π)
R
R
p
holds for any fixed but arbitrary p ∈ N0 = {0, 1, 2, . . .}.
Proof. Denote
(4.2)
g (x) = e−2πixξm f (x + xm ) ,
for any fixed but arbitrary constant xm ∈ R.
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 ) .
Denote the Fourier transform of g(p) either by F g (p) (ξ) , or F g (p) (x) (ξ) , or also as g (p) (x) ˆ (ξ) .
From this and Gasquet et al. [8, p. 155–157] we find
1
(p)
(4.4)
ξ p ĝ (ξ) =
(ξ) .
pFg
(2πi)
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
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
13
or
Z
(4.6)
(p) 2
F g (ξ) dξ =
R
Z
(p)
g (x)2 dx.
R
Finally denote
Z
(4.7)
(µ2p ) fˆ 2 =
| |
R
2
(ξ − ξm )2p fˆ (ξ) dξ
2
th
the 2p moment of ξ for fˆ for any fixed but arbitrary constant ξm ∈ R and 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
=
ξ 2p e2πixm (ξ+ξm ) fˆ (ξ + ξm ) dξ
ZR
=
ξ 2p |ĝ (ξ)|2 dξ
R
Z
(p) 2
1
F g (ξ) dξ
=
(2π)2p R
Z
(p)
1
g (x)2 dx.
=
2p
(2π)
R
From this and (4.2) we find
Z (p) 2
1
−2πixξm
(µ2p ) fˆ 2 =
e
f
(x
+
x
)
dx.
m
| |
(2π)2p R
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
completing the proof of the required identity (4.1) ([16] – [24]).
5. T HE pTH -D ERIVATIVE OF THE P RODUCT OF T WO F UNCTIONS
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
(m)
x, then the pth -derivative of the product f1 f2 is given, in terms of the lower derivatives f1 ,
(p−m)
f2
by
p X
p
(m) (p−m)
(p)
(5.1)
(f1 f2 ) =
f1 f2
m
m=0
for any fixed but arbitrary p ∈ N0 = {0} ∪ N.
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J OHN M ICHAEL R ASSIAS
(0) (0)
Proof. In fact, for p = 0 the formula (5.1) is trivial, as (f1 f2 )(0) = f1 f2 . When p = 1 the
(1)
(1)
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 p X
X
p
p
(m+1) (p−m)
(m) (p+1−m)
(p+1)
f1
f2
+
f1 f2
(f1 f2 )
=
m
m
m=0
m=0
p p+1 X
X
p
p
(m) (p+1−m)
(m) (p+1−m)
+
f1 f2
=
f1 f2
m
m−1
m=0
m=1
p X
p
p
p
p
(p+1)
(m) (p+1−m)
(p+1)
=
f1
f2 +
+
f1 f2
+
f1 f2
p
m−1
m
0
m=1
p X
p+1
p+1
p+1
(p+1)
(m) (p+1−m)
(p+1)
f2 +
f1 f2
+
f1 f2
=
f1
m
0
p+1
m=1
p+1 X p+1
(m) (p+1−m)
=
f1 f2
,
m
m=0
as the classical Pascal identity
(P)
p
m−1
+
p
m
=
p+1
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
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
p−k (k) (5.4)
fa = a f ,
k
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
R
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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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
15
From this, (4.2) and
|g (x)| = e−2πixξm f (x + xm ) = |f (x + xm )|
one gets that
Z
(µ2p )|f |2 =
(5.6)
x2p |g (x)|2 dx.
R
Besides (5.5) and |fa | = |f | yield
Z
(µ2p )|f |2 =
(5.7)
(x − xm )2p |fa (x)|2 dx.
R
6. G ENERALIZED I NTEGRAL I DENTITIES
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 thatwp(x) = (x − xm )p w(x) for any fixed but arbitrary constant xm ∈ R and v = p − 2q,
0 ≤ q ≤ p2 , then
i)
Z
Z
v−1
X
r (r)
v
(v)
(v−r−1)
(6.1)
wp (x) h (x) dx =
(−1) wp (x) h
(x) + (−1)
wp(v) (x) h (x) dx
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
wp(v) (x) h (x) dx
R
holds if the condition
(6.3)
v−1
X
(−1)r lim wp(r) (x) h(v−r−1) (x) = 0,
r=0
|x|→∞
holds, and if all these integrals exist.
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
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16
J OHN M ICHAEL R ASSIAS
"
v−1
X
= wp h(v) (x) −
!
r
v
(−1) wp(r+1) h(v−r−1)
#
Z
wp(v+1) h (x) dx
(x) + (−1)
r=0
#
" v
Z
X
v
r−1 (r) (v−(r−1)−1)
(v)
(x) − (−1)
(x) −
(−1) wp h
= wp h
= wp h(v) (x) +
r=1
v
X
!
r
(−1)
v+1
wp(r) h((v+1)−r−1)
wp(v+1) h (x) dx
Z
wp(v+1) h (x) dx
(x) + (−1)
r=1


(v+1)−1
=
X
r
(−1)
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).
ii) The proof of (6.2) is clear from (6.1) and (6.3).
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
Z
Z
2 (p−2q)
2
p−2q
(l)
wp (x) f (x)
dx = (−1)
wp(p−2q) (x) f (l) (x) dx,
(6.4)
R
R
p
for 0 ≤ l ≤ q ≤ 2 ,
ii) If
h (x) = Re rqkj f (k) (x) f (j) (x) ,
(6.5)
and if (6.3) holds, then from (6.2) we get
Z
(p−2q)
dx
wp (x) Re rqkj f (k) (x) f (j) (x)
R
Z
p−2q
= (−1)
wp(p−2q) (x) Re rqkj f (k) (x) f (j) (x) dx,
R
q− k+j
2
where rqkj = (−1)
for 0 ≤ k < j ≤ q ≤
p
, and v = p − 2q.
2
7. L AGRANGE T YPE D IFFERENTIAL I DENTITY
We state and prove the new Lagrange type differential identity.
ax
Proposition 7.1. If f : R →
√ C is a complex valued function of a real variable x, and fa = e 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
p
p
Bpkj = spk
β 2p−j−k ,
0 ≤ k < j ≤ p,
k
j
where spk = (−1)p−k (0 ≤ k ≤ p), then
(7.1)
p
X
(p) 2 X
2
fa =
Apk f (k) + 2
Bpkj Re rpkj f (k) f (j) ,
k=0
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
17
holds for any fixed but arbitrary p ∈ N0 = {0} ∪ N, where (·) is the conjugate of (·), and
k+j
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
X
|rk zj − rj zk |2 ,
0≤k<j≤p
with rk , zk ∈ C, such that 0 ≤ k ≤ p, takes the new form
2 " p
p
!
!
p
X
X 2
X
X
|rl |
|zk |2 −
|rk |2 |zj |2
rk zk =
l=0
k=0
0≤k<j≤p
k=0
#
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
+ ··· +
|rl |2 −
|rl |2 −
!
X
l=0
|rk |2
p
X
!
|rk |2
|z1 |2
k6=1
#
X
|zp |2 + 2
k6=p
Re (rk rj zk zj ),
0≤k<j≤p
or the new identity
2
p
p
X
X
X
rk zk =
|rk |2 |zk |2 + 2
Re (rk rj zk zj ),
(7.3)
k=0
k=0
0≤k<j≤p
because
|rk zj − rj zk |2 = (rk zj − rj zk ) (rk zj − rj zk )
(7.4)
= |rk |2 |zj |2 + |rj |2 |zk |2 − (rk rj zk zj + rk rj zk zj )
= |rk |2 |zj |2 + |rj |2 |zk |2 − 2 Re (rk rj zk zj ) ,
as well as
(7.5)
X
X
|rk |2 |zj |2 +
0≤k<j≤p
|rj |2 |zk |2
0≤k<j≤p
X
=
|rk |2 |zj |2
0≤k6=j≤p
!
=
X
|rk |2
!
X
|z0 |2 +
k6=0
|rk |2
!
X
|z1 |2 + · · · +
k6=1
|rk |2
|zp |2 .
k6=p
Setting
(7.6)
rk =
p
k
a
p−k
=
p
k
p
k
p−k
(−βi)
p−k
= (−i)
p
k
β p−k ,
one gets that
(7.7)
2
|rk | =
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
2
β 2(p−k) = Apk ,
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18
J OHN M ICHAEL R ASSIAS
and
p
k
p
j
(−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
k
j
= Bpkj rpkj ,
rk rj =
(7.8)
where Apk , Bpkj ∈ R, and spk = (−1)p−k ∈ {±1} as well as
rpkj = i2p−k−j = (−1)p−
k+j
2
∈ {±1, ±i} .
Thus employing (5.4) and substituting
zk = f
(k)
, rk =
p
k
ap−k
(0 ≤ k ≤ p),
in (7.3), we complete from (7.6) – (7.8), the proof of the identity (7.1).
We note that
spk = 1,
if p ≡ k(mod 2);
spk = −1, if p ≡ (k + 1) mod 2.
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);
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.
7.1. Special cases of (7.1):
(i)
(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 ,
(7.9)
because
Re (iz) = − Im (z) ,
(7.10)
where Im (z) is the imaginary part of z ∈ C.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
19
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
2
= |−iβf + f 0 |
= (−iβf + f 0 ) iβf + f 0
2
= 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 .
(ii)
(2) 2 ax 00 2
fa = (e f ) (7.11)
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).
8. O N THE H EISENBERG -PAULI -W EYL I NEQUALITY
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.
ax
Denote fa = e 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) =
e2iπξx fˆ (ξ) dξ.
R
Also we denote
Z
(µ2p )w,|f |2 =
w2 (x) (x − xm )2p |f (x)|2 dx
R
th
2
the 2p weighted moment of x for |f | with weight function w and
Z
2
(µ2p ) fˆ 2 =
(ξ − ξm )2p fˆ (ξ) dξ
| |
R
2
the 2pth moment of ξ for fˆ . Besides we denote
hpi p
p
p−q
q
Cq = (−1)
, if 0 ≤ q ≤
= the greatest integer ≤
,
q
p−q
2
2
Z
hpi
2
p−2q
Iql = (−1)
wp(p−2q) (x) f (l) (x) dx, if 0 ≤ l ≤ q ≤
,
2
R
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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20
J OHN M ICHAEL R ASSIAS
p−2q
Z
wp(p−2q)
Iqkj = (−1)
(x) Re rqkj f
(k)
(x) f
(j)
(x) dx,
R
if 0 ≤ k < j ≤ q ≤
hpi
2
,
where
rqkj = (−1)q−
k+j
2
∈ {±1, ±i}
and
wp = (x − xm )p w.
We assume that all these integrals exist. Finally we denote
Dq =
q
X
X
Aql Iql + 2
l=0
Bqkj Iqkj ,
0≤k<j≤q
if |Dq | < ∞ holds for 0 ≤ q ≤ p2 , where
2
q
Aql =
β 2(q−l) ,
l
q
q
Bqkj = sqk
β 2q−j−k ,
k
j
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)
r
(−1) lim
|x|→∞
r=0
for 0 ≤ l ≤ q ≤
wp(r)
2 (p−2q−r−1)
(l)
(x) f (x)
= 0,
p
, and
2
p−2q−1
(8.2)
X
r
(−1) lim
r=0
for 0 ≤ k < j ≤ q ≤
Pauli-Weyl inequality .
|x|→∞
wp(r)
(x) Re rqkj f
(k)
(x) f
(j)
(p−2q−r−1)
(x)
= 0,
p
. From these preliminaries we establish the following Heisenberg2
Theorem 8.1. If f ∈ L2 (R) , then
q
q
q
1 p
2p
√
(8.3)
(µ2p )w,|f |2 2p (µ2p ) fˆ 2 ≥
|Ep,f |,
| |
2π p 2
holds for any fixed but arbitrary p ∈ N.
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.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
21
kp2
2
> 0, kp ∈ R − {0}, p ∈ N, then this a-differential equation holds iff
"
#
m+1
p−1
∞
X
X
kp2 x2p
δ
m+1
j
2πixξm
f (x) = e
aj x δ 1 +
(−1)
,
2(m+1)p+j
Pp2p+j Pp4p+j · · · Pp
m=0
j=0
√
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
In fact, if cp =
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 .
0
R
2
Thus
if
f is either odd or even, then |f | is even and xm = 0. Similarly, if ξm is the mean of ξ
ˆ2
for f , then
Z ˆ 2
ξm =
ξ f (ξ) dξ.
R
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,
|x|→∞
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
=
w (x) (x − xm ) |fa (x)| dx ·
fa (x) dx .
(2π)2p
R
R
From (8.6) and the Cauchy-Schwarz inequality, we find
Z
2
1
(p)
(8.7)
Mp ≥
wp (x) fa (x) fa (x) dx ,
(2π)2p
R
where wp = (x − xm )p w, and fa = eax f .
From (8.7) and the complex inequality
(8.8)
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
|ab| ≥
1
ab + ab
2
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22
J OHN M ICHAEL R ASSIAS
(p)
with a = wp (x) fa (x), b = fa (x), we get
Z
2
1
1
(p)
(p)
wp (x) fa (x) fa (x) + fa (x)fa (x) dx .
(8.9)
Mp ≥
(2π)2p 2 R
From (8.9) and the generalized differential identity (*), one finds


 2
Z
[p/2]
p−2q X
2
1
d
(8.10)
Mp ≥ 2(p+1) 2p  wp (x) 
Cq p−2q fa(q) (x)  dx .
2
π
dx
R
q=0
From (8.10) and the Lagrange type differential identity (7.1), we find


Z
[p/2]
q
X
2
1
dp−2q X


(8.11) Mp ≥ 2(p+1) 2p
wp (x)
Cq p−2q
Aql f (l) (x)
2
π
dx
R
q=0
l=0
+2
X
Bqkj Re rqkj f (k) (x) f (j) (x)
#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 exist, one gets
Z
Z
2
2
dp−2q (l)
p−2q
wp (x) p−2q f (x) dx = (−1)
(8.12)
wp(p−2q) (x) f (l) (x) dx
dx
R
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 .
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
22(p+1) π 2p
2
Ep,f
, (Hp ),
where
[p/2]
Ep,f =
X
Cq Dq , if |Ep,f | < ∞
q=0
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 = 21 kp2 > 0, kp ∈ R − {0}, and any fixed but arbitrary
p ∈ N.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
23
We consider the general a-differential equation
dp y
(ap )
+ kp2 xpδ y = 0
p
dx
th
of p 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
dp y
2 p
(δp )
p + kp xδ y = 0,
dxδ
because
dy dxδ
dy d (x − xm )
dy
dy
=
=
=
,
dx
dxδ dx
dxδ
dx
dxδ
and
dp y
dp y
=
, p ∈ N.
dxp
dxpδ
In order to solve equation (δp ) one may employ the following powerPseries method ([18], [21],
n
[25]) in (δp ). In fact, we consider the power series expansion y = ∞
n=0 an xδ 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
∞
dp y X n
Pp an xn−p
p =
δ
dxδ
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
Ppn+2p = (n + 2p) (n + 2p − 1) (n + 2p − 2) · · · (n + p + 1))
=
=
−1
X
Ppn+2p an+2p xn+p
δ
n=−p
∞
X
+
∞
X
Ppn+2p an+2p xn+p
δ
n=0
Ppn+2p an+2p xn+p
δ
n=0
(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
(Rp )
an+2p = −
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
kp2 an
Ppn+2p
,
n ∈ N0 , p ∈ N.
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24
J OHN M ICHAEL R ASSIAS
From the “null condition”
ap = 0, ap−1 = 0, . . . , a2p−1 = 0
and (Rp ) we get
a3p = a5p = a7p = · · · = 0, a3p+1 = a5p+1 = a7p+1 = · · · = 0, . . . ,
and
a4p−1 = a6p−1 = a8p−1 = · · · = 0,
respectively. From (Rp ) and n = 2pm with m ∈ N0 one finds the following p sequences
(a2pm+2p ) , (a2pm+2p+1 ) , (a2pm+2p+2 ) , . . . , (a2pm+3p−1 ) ,
with fixed p ∈ N and all m ∈ N0 , such that
a2p = −
kp2
2p a0 ,
Pp
a4p = −
kp2
kp4
Pp
Pp2p Pp4p
4p a2p =
kp2m+2
m+1
a2pm+2p = (−1)
Pp2p Pp4p · · · Pp2pm+2p
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
a0 , . . . ,
a0
kp4
Pp2p+1 Pp4p+1
a1 , . . . ,
kp2m+2
a1
Pp2p+1 Pp4p+1 · · · Pp2pm+2p+1
are the elements of the second sequence (a2pm+2p+1 ) in terms of a1 , . . ., and
a2p+(p−1)
kp2
= a3p−1 = − 3p−1 ap−1 ,
Pp
kp2
kp2
a2p+(3p−1) = a5p−1 = − 5p−1 a3p−1 = 3p−1 5p−1 ap−1 , . . . ,
Pp
Pp Pp
kp2m+2
a2pm+(3p−1) = a(2m+3)p−1 = (−1)m+1 3p−1 5p−1
a
(2m+3)p−1 p−1
P p Pp
· · · Pp
are the elements of the last sequence (a2pm+3p−1 ) in terms of ap−1 . Therefore we find the
following p solutions
m+1
∞
X
kp2 x2p
δ
m+1
y0 = y0 (xδ ) = 1 +
(−1)
,
2(m+1)p
2p 4p
P
P
·
·
·
P
p
p
p
m=0
"
#
m+1
∞
X
kp2 x2p
δ
m+1
,...,
y1 = y1 (xδ ) = xδ 1 +
(−1)
2(m+1)p+1
Pp2p+1 Pp4p+1 · · · Pp
m=0
"
#
∞
2 2p m+1
X
k
x
p
yp−1 = yp−1 (xδ ) = xp−1
1+
(−1)m+1 3p−1 5p−1 δ
,
δ
(2m+3)p−1
P
P
·
·
·
P
p
p
p
m=0
or equivalently the
"
#
∞
2 2p m+1
X
x
k
p
yj = yj (xδ ) = xjδ 1 +
(−1)m+1 2p+j 4p+j δ
2(m+1)p+j
Pp Pp
· · · Pp
m=0
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
25
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
a0 = 0, a1 = 0, a2 = 0, . . . , ap−2 = 0, ap−1 = 1,
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;
······························ ;
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 = 0 is
W (Yp (0)) = y0 (0)
y00 (0)
···
(p−1)
y0
(0)
y1 (0)
y10 (0)
···
(p−1)
y1
(0)
···
···
···
···
yp−1 (0)
y 0 p−1 (0)
···
(p−1)
yp−1 (0)
=
1 0 ··· 0 0 1 ··· 0 ..
. . .. = 1 6= 0,
. . .
0 0 ··· 1 yielding that these p partial solutions y0 , y1 , . . . , yp−1 of (δp ) are linearly independent. Thus the
P
above formula y = fa (x) = p−1
j=0 aj yj gives the general solution of the equation (δp ) (and also
of (ap )).
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 )
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
dy
+ k12 xδ y = 0,
dx
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26
J OHN M ICHAEL R ASSIAS
where xδ = x − xm 6= 0, y = fa (x) = eax f (x), a = −2πξm i, is satisfied by
y0 = y0 (x)
∞
X
=1+
(−1)m+1
=1+
m=0
∞
X
m+1
(k12 x2δ )
2(m+1)
P12 P14 · · · P1
m+1
m+1
(−1)
m=0
(c1 x2δ )
(m + 1)!
(by the power series method and because
2(m+1)
P12 P14 · · · P1
or
y0 = 1 +
P∞
m tm
m=1 (−1) m!
∞
X
= 2 · 4 · · · · 2 (m + 1) = 2m+1 (m + 1)!),
m
(−1)m
m=1
−t
(cx2δ )
m!
2
2
= 1 + e−cxδ − 1 = e−cxδ
(because
= e − 1).
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
1
2
(H2 )
(µ4 )|f |2 · (µ4 ) fˆ 2 ≥
E2,f
,
4
| |
64π
holds, if
Z
x4δ |f (x)|2 dx
(µ4 )|f |2 =
R
and
Z
2
(µ4 ) fˆ 2 =
ξδ4 fˆ (ξ) dξ
| |
R
with xδ = x − xm , and ξδ = ξ − ξm , are the fourth order moments, and
Z
Z
√
−2iπξx
fˆ (ξ) =
e
f (x)dx, f (x) =
e2iπξx fˆ (ξ)dξ,
i = −1,
R
R
as well as
E2,f = 2
Z h
1−
2 2
4π 2 ξm
xδ
2
|f (x)| −
x2δ
0
2
|f (x)|
−4πξm x2δ
Im f
(x) f 0
i
(x) dx,
R
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 = 21 k22 > 0, k2 ∈ R − {0},
xδ = x − xm 6= 0, or equivalently
(a2 )
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
d2 y
+ k22 x2δ y = 0
2
dx
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
27
holds iff
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
d2 y
+ k22 x2δ y = 0
dx2δ
(δ2 )
is equivalent to the following Bessel equation
"
2 #
2
du
1
2d u
2
z
+z
+ z −
u=0
2
dz
dz
4
of order r =
1
,
4
with u = √y
|xδ |
and z =
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
for some constants c20 , c21 ∈ C.
In fact, if we denote
(
S = sgn (xδ ) =
(8.15)
1,
xδ > 0
,
−1, xδ < 0
then one gets
du
du
=
dz
dxδ
dz
=
dxδ
"
S
dy
1 y
−
3/2 dx
2 |xδ |5/2
δ
|xδ |
#,
|k2 | ,
and
d2 u
d
=
2
dz
dxδ
du
dz
Thus we establish
z2
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 .
2
d2 u
du
1
1
3/2 d y
+
z
=
|x
|
+
u.
δ
2
dz 2
dz
4
dxδ 16
But
d2 y
= −k22 x2δ y,
2
dxδ
or
|xδ |3/2
d2 y
y
= −4z 2 u.
= −k22 x4δ p
2
dxδ
|xδ |
Therefore the above-mentioned Bessel equation holds.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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28
J OHN M ICHAEL R ASSIAS
However,
dy
dy dxδ
dy d (x − xm )
dy
=
=
=
,
dx
dxδ dx
dxδ
dx
dxδ
and
dy dxδ
d2 y
d dy
d
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
2
2
ax
y = fa (x) = e f (x) = |xδ | c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
2
2
or
p
1
1
−ax
2
2
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
f (x) = |xδ |e
2
2
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
1
2 2p
k22 5
4
2
|xδ |J1/4
|k2 | xδ = S 1 |k2 | xδ −
x + ··· ,
2
4·5 δ
Γ 4
because
5
1
1
9
5
1
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
Therefore
p
|xδ |J−1/4
1
|k2 | x2δ
2
=
Γ
1
4
π
1
k22 4
p
1−
x + ··· ,
4
3·4 δ
|k2 |
because
√
1
3
1
1
π
Γ
Γ
=Γ
Γ 1−
=
2,
=
π
4
4
4
4
sin 14 π
or
√
√
3
7
3
3
3
3 2π
π 2
, . . ..
, and Γ
=Γ 1+
= Γ
=
Γ
=
4
4
4
4
4
4Γ 14
Γ 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 arbitrary constants a0 = c20 and
∞
P
a1 = c21 such that y =
an xnδ , about xδ0 = 0, converging (absolutely) in
n=0
an = ∞,
|xδ | < ρ = lim n→∞ an+1 Thus
∞
xδ 6= 0.
∞
X
X
dy
=
nan xn−1
=
(n + 1) an+1 xnδ ,
δ
dxδ
n=1
n=0
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
29
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 + 4) (n + 3) an+4 xn+2
δ
n+2=0 or
n=−2
= (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)
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 +
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
y1 = y1 (xδ )
"
= xδ 1 +
∞
X
m=0
2
k2 5
xδ
m+1
(−1)m+1
(k22 x4δ )
4 · 5 · · · (4m + 4) (4m + 5)
#
k24
k26
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
= xδ −
+
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
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
1
2
|k2 | x2δ
i
,
S ∈ { ± 1},
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30
J OHN M ICHAEL R ASSIAS
and
h
SΓ
y1 =
1
4
p
|xδ |J1/4 12 |k2 | x2δ
√ p
2 2 4 |k2 |
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
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
y0 (0)
y1 (0)
W (y0 , y1 ) (0) = y00 (0)
y10 (0)
1
=
0
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 5
k22 4
= xδ − xδ + · · ·
1 − xδ + · · ·
20
12
2
k
= xδ + 2 x5δ + · · · ,
30
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
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
31
and
Z
(µ6 ) fˆ 2 =
| |
R
2
ξδ6 fˆ (ξ) dξ
with xδ = x − xm , and ξδ = ξ − ξm , and
fˆ (ξ) =
Z
−2iπξx
e
Z
f (x)dx, f (x) =
R
e2iπξx fˆ (ξ)dξ,
R
as well as
E3,f = −3
Z h
i
2
2 2
2 1 − 6π 2 ξm
xδ |f (x)|2 − 3x2δ |f 0 (x)| −12πξm x2δ Im f (x) f 0 (x) dx,
R
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
√
k2
holds, for a = −2πξm i, i = −1, fa = eax f , and a constant c3 = 23 > 0, k3 ∈ R, or
equivalently iff
2πixξm
f (x) = e
2
X
j=0
"
aj xjδ
1+
∞
X
(−1)m+1
m=0
m+1
(k32 x6δ )
×
(4 + j) (5 + j) (6 + j) · · · (6m + 4 + j) (6m + 5 + j) (6m + 6 + j)
#
holds, where xδ 6= 0, and aj (j = 0, 1, 2) are arbitrary constants in C.
Consider the a-differential equation
d3 y
+ k32 x3δ y = 0,
dx3
(a3 )
with y = fa (x) and the equivalent δ-differential equation
d3 y
+ k32 x3δ y = 0,
3
dxδ
(δ3 )
with xδ = x − xm and k3 ∈ R − {0}, such that d3 y/ dx3 = d3 y/ dx3δ .
the power series method for (δ3 ), one considers the power series expansion y =
PEmploying
∞
n
n=0 an xδ about xδ0 = 0, converging (absolutely) in
an = ∞,
|xδ | < ρ = lim n→∞ an+1 xδ 6= 0.
Thus
k32 x3δ y
=
∞
X
k32 an xn+3
,
δ
n=0
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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32
J OHN M ICHAEL R ASSIAS
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
n=−3
= 6a3 + 24a4 xδ +
60a5 x2δ
+
∞
X
P3n+6 an+6 xn+3
δ
n=0
=
∞
X
P3n+6 an+6 xn+3
δ
n=0
(with a3 = a4 = a5 = 0), where P3n+6 = (n + 6) (n + 5) (n + 4). Therefore from these and
equation (δ3 ) one gets the recursive relation
an+6 = −
(R3 )
k32 an
,
P3n+6
n ∈ N0
From “the null condition”
(N3 )
a3 = 0,
a4 = 0,
and
a5 = 0
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 = −
a6m+6 = (−1)m+1
k32m+2
a0 ,
4 · 5 · 6 · · · · · (6m + 4) (6m + 5) (6m + 6)
and
k32
a1 ,
5·6·7
k32
k34
a13 = −
a7 =
a1 ,
11 · 12 · 13
5 · 6 · 7 · 11 · 12 · 13
...,
a7 = −
a6m+7 = (−1)m+1
k32m+2
a1 ,
5 · 6 · 7 · · · · · (6m + 5) (6m + 6) (6m + 7)
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
33
as well as
k32
a2 ,
6·7·8
k34
k32
a14 = −
a8 =
a2 ,
12 · 13 · 14
6 · 7 · 8 · 12 · 13 · 14
...,
a8 = −
m+1
a6m+8 = (−1)
k32m+2
a2 .
6 · 7 · 8 · · · · · (6m + 6) (6m + 7) (6m + 8)
Therefore we find the following three solutions
y0 = y0 (xδ )
∞
X
=1+
(−1)m+1
m+1
(k32 x6δ )
4 · 5 · 6 · · · · · (6m + 4) (6m + 5) (6m + 6)
m=0
∞ X
a6m+6
x6m+6
=1+
δ
a
0
m=0
for a0 6= 0,
y1 = y1 (xδ )
"
= xδ 1 +
∞
X
(−1)m+1
m=0
= xδ +
∞ X
a6m+7
m=0
a1
m+1
#
m+1
#
(k32 x6δ )
5 · 6 · 7 · · · · · (6m + 5) (6m + 6) (6m + 7)
x6m+7
δ
for a1 6= 0, and
y2 = y2 (xδ )
"
= x2δ 1 +
∞
X
(−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
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,
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
a1 = 0,
a1 = 1,
a1 = 0,
a2 = 0;
a2 = 0; and
a2 = 1,
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34
J OHN M ICHAEL R ASSIAS
one gets that yj (j = 0, 1, 2) are partial solutions of (δ3 ), satisfying the initial conditions
y0 (0) = 1,
y00 (0) = 0,
y000 (0) = 0;
y1 (0) = 0,
y10 (0) = 1,
y100 (0) = 0;
y2 (0) = 0,
y20
y200
(0) = 0,
and
(0) = 1.
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
y0 (0)
y100 (0)
y200 (0)
1
0
0 1
0 = 1 6= 0,
= 0
0
0
1 yielding that these p = 3 partial solutions yj (j = 0, 1, 2) of (δ3 ) are linearly independent.
P
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 ):
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
2
E4,f
,
(H4 )
(µ8 )|f |2 · (µ8 ) fˆ 2 ≥
8
| |
1024π
holds, if
Z
x8δ |f (x)|2 dx
(µ8 )|f |2 =
R
and
Z
2
ξδ8 fˆ(ξ) dξ
(µ8 ) fˆ 2 =
| |
R
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
2 2
4 4
4 3 − 24π 2 ξm
xδ + 16π 4 ξm
xδ |f (x)|2
R
i
2
2
2 2
−8x2δ 3 − 2π 2 ξm
xδ |f 0 (x)| + x4δ |f 00 (x)|
h
2
2 2 2
0
−8πξm xδ 4 3 − π ξm xδ Im f (x) f (x)
io
dx,
+πξm x2δ Re f (x) f 00 (x) − x2δ Im f 0 (x) f 00 (x)
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)
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
35
√
of fourth order holds, for a = −2πξm i, i = −1, fa = eax f , and a constant c4 =
k4 ∈ R, or equivalently iff
"
3
∞
X
X
f (x) = e2πixξm
aj xjδ 1 +
(−1)m+1
k42
2
> 0,
m=0
j=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.
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
[C0 (A00 I00 )]2
16π 2
1 2
I
16π 2 00
Z
2
1
(1)
2
w1 (x) |f (x)| dx
16π 2
R
1
E2 ,
16π 2 1,f
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)
2
=−
w1 |f | (x) dx
ZR
= − |f (x)|2 dx
R
= −E|f |2
Z 2
= −1 = −E fˆ 2 = − fˆ (ξ) dξ
| |
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 density |f |2 and the variance (µ2 ) fˆ 2 of ξ for
| |
2
E2 2
ˆ
|f |
the probability density f is at least 16π
2 , which is the second order moment Heisenberg
1
, for E|f |2 = 1, can
Inequality (H1 ) in our Introduction. The Heisenberg lower bound H ∗ = 4π
ˆ
be different if one chooses a different formula for the Fourier transform f of f . Finally, the
above inequality (8.16) generalizes (H1 ) of our Introduction (there w = 1).
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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36
J OHN M ICHAEL R ASSIAS
(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
4
64π
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
−2 w2 (x) β |f | + |f | + 2β Im f f¯0 (x) dx ,
R
because Re if f¯0 (x) = − Im f f¯0 (x), and
Z
(2)
w2 (x) |f (x)|2 dx,
I00 =
ZR
I10 =
w2 (x) |f (x)|2 dx,
ZR
2
I11 =
w2 (x) |f 0 (x)| dx,
R
and
Z
I101 =
w2 (x) Re if f¯0 (x) dx,
R
with w2 (x) = w(x)(x − xm )2 .
It is clear that (8.17) is equivalent to
Z 1
(2)
2
(8.18)
M2 ≥
w2 − 2β w2 (x) |f (x)|2 dx
4
64π
R
2
Z
Z
2
0
−2 w2 (x) |f (x)| dx − 4β w2 (x) Im f f¯0 (x) dx
R
R
1
=
E2 ,
64π 4 2,f
or
p
4
(8.19)
1
M2 ≥ √
2π 2
q
|E2,f |,
where
(8.20)
E2,f = C0 D0 + C1 D1
= D0 − 2D1
= I00 − 2 β 2 I10 + I11 − 2βI101
Z h
i
2
(2)
=
w2 − 2β 2 w2 |f |2 − 2w2 |f 0 | − 4βw2 Im f f¯0 (x) dx,
R
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
37
√
if |E2,f | < ∞ holds. We note that if |E2,f | = 21 holds, then from (8.19) one gets 4 M2 ≥
1
(= H ∗ ),while if |E2,f | = 1, then
4π
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 | > 12 ; the same
with H ∗ if |E2,f | = 21 ; and smaller than H ∗ , if 0 < |E2,f | < 12 . 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
6
256π
1
=
[C0 (A00 I00 ) + C1 (A10 I10 + A11 I11 + 2B101 I101 )]2
6
256π
2
1 =
I00 − 3 β 2 I10 + I11 − 2βI101
6
256π Z
1
(3)
=
− w3 (x) |f (x)|2 dx
256π 6
R
2
Z
h
i
(1)
2
2
0 2
0
+ 3 w3 (x) β |f | + |f | + 2β Im f f (x) dx ,
≥
R
because
Z
I00 = −
(3)
w3
Z
2
(x) |f (x)| dx,
(1)
w3 (x) |f (x)|2 dx,
I10 = −
R
R
Z
I11 = −
2
(1)
w3 (x) |f 0 (x)| dx,
R
and
Z
(1)
w3
(x) f 0
(x) dx
I101 = −
(x) Re if
Z R
(1)
=
w3 (x) Im f (x) f 0 (x) dx,
R
3
with w3 (x) = w(x)(x − xm ) .
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
6
256π
R
2
Z
Z
2
(1)
(1)
0
+3 w3 |f (x)| dx + 6β w3 (x) Im f f¯0 (x) dx
R
=
R
1
2
E3,f
,
6
256π
or
(8.23)
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
p
6
1
M3 ≥ √
2π 3 2
q
3
|E3,f |
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38
J OHN M ICHAEL R ASSIAS
where
E3,f =
Z h
(3)
(1)
−w3 + 3β 2 w3
2
(1)
(1)
|f |2 + 3w3 |f 0 | + 6βw3 Im f f¯0
i
(x) dx,
R
√
1
if |E3,f | < ∞ holds. We note that if |E3,f | = 41 , then from (8.23) we find 6 M3 ≥ 4π
(= H ∗ ),
√
while if |E3,f | = 1, then 6 M3 ≥ 2π1√
(> H ∗ ). Thus we observe that the lower bound of
3
2
√
6
M3 is greater than H ∗ if |E3,f | > 14 ; the same with H ∗ if |E3,f | = 14 ; and smaller than H ∗ ,
if 0 < |E3,f | < 41 . 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
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 2
=
I
−
4
β
I
+
I
−
2βI
00
10
11
101
1024π 8
2
+2 β 4 I20 + 4β 2 I21 + I22 + 4β 3 I201 + 2β 2 I202 − 4βI212
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
−2β Re f f 00 + 4β Im f 0 f 00 (x) dx
1
=
E2 ,
1024π 8 4,f
2
because
Z
I00 =
ZR
I10 =
ZR
I11 =
ZR
I20 =
ZR
I21 =
ZR
I22 =
(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,
R
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
39
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
0
I201 =
w4 (x) Re −if (x) f (x) dx =
w4 (x) Im f (x) f 0 (x) dx,
Z R
ZR
I202 =
w4 (x) Re −f (x) f 00 (x) dx = − w4 (x) Re f (x) f 00 (x) dx,
ZR
ZR
I212 =
w4 (x) Re if 0 (x) f 00 (x) dx = − w4 (x) Im f 0 (x) f 00 (x) dx,
R
R
4
with w4 (x) = w(x)(x − xm ) .
It is clear that (8.24) is equivalent to
p
8
(8.25)
M4 ≥
1
√
2π 4 2
q
4
|E4,f |,
where
Z h
(4)
w4
(2)
w4
2
(2)
−w4
2
2
− 4β
+ 2β w4 |f | + 4
+ 2β w4 |f 0 | + 2w4 |f 00 |
R
i
(2)
−8β w4 − β 2 w4 Im f f 0 − 4β 2 w4 Re f f 00 + 8βw4 Im f 0 f 00 (x) dx,
√
1
if |E4,f | < ∞ holds.We note that if |E4,f | = 18 , then from (8.25) we find 8 M4 ≥ 4π
(= H ∗ ),
√
while if |E4,f | = 1, then 8 M4 ≥ 2π1√
(> H ∗ ). Thus we observe that the lower bound of
4
2
√
8
M4 is greater than H ∗ if |E4,f | > 18 ; the same with H ∗ if |E4,f | = 18 ; and smaller than H ∗ ,
if 0 < |E4,f | < 81 . Finally, the above inequality (8.24) generalizes (H4 ) of our Section 8 (there
w = 1).
E4,f =
2
4
2
8.5. First form of (8.3), if w = 1, xm = 0 and ξm = 0. We note that β = 2πξm = 0,
(p)
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 ,
R
E2,f = 2
(8.27)
Z h
i
2
|f (x)|2 − x2 |f 0 (x)| dx,
R
(8.28)
E3,f = −3
Z h
i
2
2 |f (x)|2 − 3x2 |f 0 (x)| dx,
R
and
(8.29)
E4,f = 2
Z h
2
2
0
2
4
00
2
i
12 |f (x)| − 24x |f (x)| + x |f (x)| dx,
R
respectively, if |Ep,f | < ∞ holds for p = 1, 2, 3, 4.
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.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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40
J OHN M ICHAEL R ASSIAS
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
It is obvious, if β = 0, that
q−k
Bqkj = (−1)
(8.31)
q
k
q
j
β 2q−j−k = 0,
0 ≤ k < j ≤ q,
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
(8.32)
Dq = Aqq Iqq = Iqq = (−1)
wp(p−2q) (x) f (q) (x) dx,
R
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
wp(p−2q) (x) = (xp )(p−2q) = p (p − 1) · · · (p − (p − 2q) + 1) xp−(p−2q) ,
or
wp(p−2q) (x) =
(8.33)
p! 2q
p!
x2q =
x ,
(p − (p − 2q))!
(2q)!
0≤q≤
hpi
2
.
From (8.32) and (8.33) we get the formula
p−2q
Dq = (−1)
(8.34)
p!
(2q)!
Z
2
x2q f (q) (x) dx,
R
if |Dq | < ∞ holds for 0 ≤ q ≤ p2 .
Therefore from (8.34) one finds that
[p/2]
Ep,f =
X
Cq Dq
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)
Ep,f =
Z [p/2]
X
p−q
(−1)
R q=0
if |Ep,f | < ∞ holds for 0 ≤ q ≤
Let
p
p!
p − q (2q)!
p−q
q
2
x2q f (q) (x) dx,
p
, when w = 1 and xm = ξm = 0.
2
Z
(8.36)
(m2p )|f |2 =
x2p |f (x)|2 dx
R
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
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
41
2
the 2p moment of ξ for fˆ about the origin ξm = 0. Denote
hpi
p!
p
p−q
p−q
(8.38)
εp,q = (−1)
, if p ∈ N and 0 ≤ q ≤
.
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,
th
R q=0
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.
| |
2
If f ∈ L (R), and all the above assumptions hold, then
v
u
[p/2]
u
q
q
X
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
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 .
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)!
= −120,
5
5!
5−1
5−1
= 300,
ε5,1 = (−1)
1
5 − 1 (2 · 1)!
and
5−2
ε5,2 = (−1)
5
5!
5 − 2 (2 · 2)!
5−2
2
= −25.
Therefore
(8.42)
E5,f = −5
Z h
2
2
0
2
4
00
2
i
24 |f (x)| − 60x |f (x)| + 5x |f (x)| dx,
R
if |E5,f | < ∞ holds.
Similarly if p = 6, then p2 = 3; q = 0, 1, 2, 3. Thus from (8.39) one finds
ε6,0 = 720,
ε6,1 = −2160,
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
ε6,2 = 270,
and
ε6,3 = −2.
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42
J OHN M ICHAEL R ASSIAS
Therefore
(8.43)
E6,f = 2
Z h
i
2
2
2
360 |f (x)|2 − 1080x2 |f 0 (x)| + 135x4 |f 00 (x)| − x6 |f 000 (x)| dx,
R
if |E6,f | < ∞ holds. In the same way one gets
Z h
i
2
2
2
2
2
0
4
00
6
000
(8.44) E7,f = −7
720 |f (x)| − 2520x |f (x)| + 420x |f (x)| − 7x |f (x)| dx,
R
(8.45) E8,f = 2
Z h
2
2
20160 |f (x)|2 − 80640x2 |f 0 (x)| + 16800x4 |f 00 (x)|
R
2 i
2
−448x6 |f 000 (x)| + x8 f (4) (x) dx,
and
(8.46) E9,f = −9
Z h
2
2
40320 |f (x)|2 − 181440x2 |f 0 (x)| + 45360x4 |f 00 (x)|
R
2 i
2
−1680x6 |f 000 (x)| + 9x8 f (4) (x) dx,
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
(w (x) xpδ )(p−2q) f (q) (x) dx
Dq = Iqq = (−1)
R
Z p−2q
X p − 2q 2
p−2q
w(m) (x) (xpδ )(p−2q−m) f (q) (x) dx
= (−1)
m
R m=0
Z p−2q
X p − 2q 2
p!
p−(p−2q−m) (q)
p−2q
= (−1)
w(m) (x)
xδ
f (x) dx,
m
(p − (p − 2q − m))!
R
m=0
or
(8.47)
Z "p−2q
X
Dq = (−1)p−2q
R
m=0
p!
(2q + m)!
p − 2q
m
#
(q)
2
w(m) (x) xm
x2q
(x) dx,
δ
δ f
if |Dq | < ∞ holds for 0 ≤ q ≤ p2 .
If m = 0, then one finds from (8.47) the formula (8.34). Therefore from (8.47) one gets that
(8.48) Ep,f =
Z [p/2]
X
p−q
(−1)
R q=0
×
p
p−q
"p−2q
X
m=0
p−q
q
p!
(2q + m)!
p − 2q
m
#
(q)
2
w(m) (x) xm
x2q
(x) dx,
δ
δ f
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).
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
43
If we denote
(8.49)
p−q
εp,q,w (x) = (−1)
p
p−q
p−q
q
"p−2q
X
m=0
p!
(2q + m)!
p − 2q
m
#
w
(m)
(x) xm
δ
,
then one gets from (8.48) that
Z [p/2]
[p/2] Z
X
X
2
(q)
2
2q (q)
(8.50)
Ep,f =
εp,q,w (x)xδ f (x) dx =
εp,q,w (x) x2q
(x) dx,
δ f
R q=0
q=0
R
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 .
It is clear that the formula
[p/2]
p
p−q
(8.51) Ep,f =
(−1)
q
p−q
q=0
#
Z "p−2q
X
2
p!
p − 2q
2q (q)
dx,
×
w(m) (x) xm
x
f
(x)
δ
δ
m
(2q
+
m)!
R m=0
if |Ep,f | < ∞ holds for 0 ≤ q ≤ p2 .
Therefore from (8.3) and (8.49) – (8.50) we get the following Corollary 8.3.
X
p−q
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
 1

[p/2] Z
p
q
X
q
2 1 
2q (q)
2p
√
(8.52)
εp,q,w (x) xδ f (x) dx ,
(µ2p )w,|f |2 2p (m2p ) fˆ 2 ≥
| |
2π p 2 R
q=0
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 ,
and
ε2,1,w (x) = −2w (x) .
Therefore from (8.48) one obtains
Z h
i
2
(8.53) E2,f =
2w (x) + 4w0 (x) xδ + w00 (x) x2δ |f (x)|2 − 2w (x) x2δ |f 0 (x)| dx,
R
if |E2,f | < ∞.
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J OHN M ICHAEL R ASSIAS
This result (8.53) can be found also from (8.20), where β = 2πξm = 0 and thus
Z h
i
2
(2)
2
0
(8.54)
E2,f =
w2 (x) |f (x)| − 2w2 (x) |f (x)| dx,
R
if |E2,f | < ∞ holds, with
w2 (x) = w (x) (x − xm )2 = w (x) x2δ ,
(1)
w2 (x) = 2w (x) xδ + w0 (x) x2δ
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) =
p!
xp−r ,
(p − r)! δ
and
p! 2q
x .
(2q)! δ
of the above theorem take the form
wp(p−2q) =
Therefore the integrals Iql , Iqkj
Iql = (−1)p−2q
(8.55)
p!
(µ2q ) f (l) 2 ,
| |
(2q)!
0≤l≤q≤
hpi
,
2
and
(8.56)
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
(k)
(j)
(8.57)
fkj (x) = Re rqkj f (x) f (x) , 0 ≤ k < j ≤ q ≤
,
2
with rqkj = (−1)q−
k+j
2
, and
Z
x2q
δ
(l)
f (x)2 dx,
(µ2q ) f (l) 2 =
| |
R
2
is the 2q th moment of x for f (l) , and
Z
(8.59)
(µ2q )fkj =
x2q
δ fkj (x)dx,
(8.58)
hpi
0≤l≤q≤
0≤k<j≤q≤
2
hpi
R
2
,
,
is the 2q th moment of x for fkj .
We note that if 0 ≤ k = j = l ≤ q ≤ p2 , then
2
(8.60)
fll (x) = sql f (l) (x) ,
and thus
(8.61)
hpi
,
(µ2q )fll = sql (µ2q ) f (l) 2 , 0 ≤ l ≤ q ≤
| |
2
where sql = (−1)q−l .
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
45
We consider Aql and Bqkj and β as in the theorem, and εp,q as in (8.38). Therefore
" q
#
X
X
p!
(8.62)
Dq = (−1)p−2q
Aql (µ2q ) f (l) 2 + 2
Bqkj (µ2q )fkj ,
| |
(2q)!
l=0
0≤k<j≤q
if |Dq | < ∞ holds for 0 ≤ q ≤ p2 . From (8.55) – (8.62) one gets
" q
#
[p/2]
X X
X
(8.63)
Ep,f =
Aql (µ2q ) f (l) 2 + 2
Bqkj (µ2q )fkj ,
| |
q=0
l=0
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,
R
and
Z
(µ2p ) fˆ 2 =
| |
ξδ2p
R
ˆ 2
f (ξ) dξ
2
be the 2p moment of x for |f | , and the 2p 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
2p
Up =
(µ2p )|f |2 2p (µ2p ) fˆ 2 .
| |
th
2
th
If f ∈ L2 (R), and all the above assumptions hold, then
(8.64)

" q
# p1
[p/2]
X
X
X
Up ≥ Hp∗ 
εp,q
Aql (µ2q ) f (l) 2 + 2
Bqkj (µ2q )fkj 
| |
q=0
l=0
0≤k<j≤q
√
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
(k) (j)
(8.65)
fkj = f f
Re (rqkj ) , for 0 ≤ k < j ≤ q ≤
,
2
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46
J OHN M ICHAEL R ASSIAS
where rqkj ∈ {±1, ±i}, such that

 1,
rqkj =
 −1,

 i,
rqkj =
 −i,
if 2q ≡ (k + j) (mod 4)
; and
if 2q ≡ (k + j + 2) (mod 4)
if 2q ≡ (k + j + 1) (mod 4)
if 2q ≡ (k + j + 3) (mod 4) .
Thus


1,
if




−1, if
(8.66)
fkj = f (k) f (j)




 0,
if
p
for 0 ≤ k < j ≤ q ≤ 2 .
Therefore


1,




−1,
(8.67)
(µ2q )fkj = (µ2q )f (k) f (j)




 0,
2q ≡ (k + j) (mod 4)
2q ≡ (k + j + 2) (mod 4)
,
2q ≡ (k + j + 1) (or (k + j + 3)) (mod 4)
if 2q ≡ (k + j) (mod 4)
if 2q ≡ (k + j + 2) (mod 4)
,
if 2q ≡ (k + j + 1) (or (k + j + 3)) (mod 4)
where
Z
(µ2q )f (k) f (j) =
x2q
f (k) f (j) (x) dx,
δ
R
for 0 ≤ k < j ≤ q ≤ p2 .
Similarly if f (k) f (j) : R → R, for 0 ≤ k < j ≤ q ≤ p2 , are real valued functions of a real
variable x, we get analogous results.
9. G AUSSIAN F UNCTION
Consider w = 1, xm and ξm means and the Gaussian function f : R → C, such that
2
f (x) = c0 e−cx , where c0 , c are constants and c0 ∈ C, c > 0. It is easy to prove the integral
formula
Z
1
Γ
p
+
2
2
(9.1)
x2p e−2cx dx =
, c > 0,
p+ 12
R
(2c)
for all p ∈ N and p = 0, where Γ is the Euler gamma function [21], such that
1
1 · 3 · · · · · (2p − 1) √
Γ p+
=
π
2
2p
√
for p ∈ N and Γ 21 = π for p = 0. Note that the mean xm of x for |f |2 is given by
Z
xm =
x |f (x)|2 dx = 0.
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
ˆ
(9.2)
f (ξ) = c0
e c , c0 ∈ C, c > 0.
c
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
47
2
In this case the mean ξm of ξ for fˆ is given by
Z ˆ 2
ξm =
ξ f (ξ) dξ = 0.
R
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
(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
4 π
2p −2cx2
2p −2c∗ ξ 2
∗
= |c0 |
x e
dx
ξ e
dξ
where c =
c
c
R
R
4
4
1
1
|c0 | Γ p + 2 Γ p + 2
1 |c0 |
∗ 2p
=π
2Γ2 p +
,
1
1 = Hp
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 note that
p
2
Z
(9.4)
E|f |2 =
|f (x)|2 dx
R
Z
2
2
= |c0 |
e−2cx dx
R
2 |c0 |
1
=√ Γ
2
2c
r
√
π
1
2
= |c0 |
, where Γ
= π.
2c
2
(9.3)
If we denote
(2p − 1)!! = 1 · 3 · 5 · · · · · (2p − 1) , 0!! = (−1)!! = 1,
for p ∈ N and p = 0, respectively, then
√
1
(2p − 1)!! √
1
(9.5)
Γ p+
=
π
for
p
∈
N,
and
Γ
=
π for p = 0.
2
2p
2
We consider the Legendre duplication formula for Γ ([18], [21])
1
22p−1
(9.6)
Γ (2p) = √ Γ (p) Γ p +
, p∈N
2
π
and the factorial formula
(9.7)
Γ (p + 1) = p!, p ∈ N0 = N ∪ {0}.
We take the Hermite polynomial ([18], [21])
(9.8) Hq (x) = (2x)q −
q (q − 1)
q (q − 1) (q − 2) (q − 3)
(2x)q−2 +
(2x)q−4
1!
2!
q
q
q!
− · · · + (−1)[ 2 ] q (2x)q−2[ 2 ] ,
!
2
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q ∈ N0 ,
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48
where
[21])
J OHN M ICHAEL R ASSIAS
q
2
=
q
2
if q is even and
q
2
=
q−1
2
if q is odd. We consider the Rodrigues formula ([18],
Hq (x) = (−1)q ex
(9.9)
2
√
dq −x2 e
, q ∈ N0 .
dxq
cx on x into (9.8) and employs
d
d
dx
1 d
√
(·) ,
(9.10)
(·) =
(·) √
=√
dx
c dx
d ( cx)
d ( cx)
then he proves the generalized Rodrigues formula
q √ q
2 d
−cx2
cx = (−1)q c− 2 ecx
e
, c > 0, q ∈ N0 .
(9.11)
Hq
dxq
2
In this paper we have 0 ≤ q ≤ p2 , p ∈ N. From (9.11) with f (x) = c0 e−cx , c0 ∈ C, c > 0,
we get
√ q
dq
(9.12)
f (x) = (−1)q c 2 f (x) Hq
cx ,
q
dx
and thus the moment
Z
2
(9.13)
(m2q ) f (q) 2 =
x2q f (q) (x) dx
| |
ZR
√ 2
q
cx dx
=
x2q (−1)q c 2 f (x) Hq
R
Z
√ 2
2 2 q
= |c0 | c
x2q e−2cx Hq
cx dx.
R
√
Substituting y = cx, c > 0 into (9.13) one gets
Z
|c0 |2
2
√
(9.14)
(m2q ) f (q) 2 =
y 2q e−2y Hq2 (y) dy.
| |
c R
We consider the Hermite polynomial
If one places
(9.15)
Hq (y) =
[ 2q ]
X
(−1)k
k=0
hpi
q!
(2y)q−2k , 0 ≤ q ≤
,
k! (q − 2k)!
2
and the Lagrange identity of the second form (7.3). Setting
q!
(2k)!
rk = (−1)
2q−2k = (−1)k
k! (q − 2k)!
k!
k
with
(2k)!
k!
q
2k
=
q
2k
2q−2k
q!
,
k! (q − 2k)!
and denoting
A∗qk
=
(2k)!
k!
2 q
2k
2
22(q−2k) ∈ R,
∗
rqkj
= 4q−(k+j) ∈ R,
s∗qkj = (−1)k+j ∈ R,
and
∗
Bqkj
=
(2k)! (2j)!
s∗qkj
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
k!j!
q
2k
q
2j
∈ R,
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
49
∗
∗
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
q!
zk zj = y 2(q−k−j) , rk zk = (−1)k
(2y)q−2k ,
k! (q − 2k)!
and

2
[q/2]
[q/2]
X
X
X
2
∗
∗


rk zk
=
A∗qk y 2q−4k + 2
(9.16)
Hq (y) =
rqkj
Bqkj
y 2(q−k−j) ,
k=0
k=0
0≤k<j≤[ 2q ]
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
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]
From (9.1) and (9.5) we find
r
r
(4 (q − k) − 1)!! π
(2 (2q − k − j) − 1)!! π
∗
(9.18)
=
, and Jqkj =
.
16q−k
2
42q−k−j
2
From (9.18) one gets
r 2 2
1
π
(2k)!
q
(9.19)
Ãqk = 2q
(4q − 4k − 1)!!,
2
2 2k
k!
r π
(2k)! (2j)!
q
q
k+j 1
(4q − 2k − 2j − 1)!!,
(9.20)
B̃qkj = (−1)
2q
2j
2
2 2k
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
1
22q−k−j
(9.22)
(4q − 2k − 2j − 1)!! = √ Γ 2q − k − j +
.
2
π
Also from (9.6) – (9.7) we get
1
(2p)!
22p
(9.23)
= √ Γ p+
, p ∈ N.
p!
2
π
Therefore from (9.19) – (9.22) and placing k, j on p into (9.23) we find
2 1
1 2k
1
q
2
Γ k+
(9.24)
Ãqk = √ 2
Γ 2q − 2k +
,
2k
2
2
π 2
∗
Jqk
(9.25) B̃qkj
1
= √ (−1)k+j 2k+j
π 2
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
q
2k
q
2j
1
1
1
×Γ k+
Γ j+
Γ 2q − k − j +
,
2
2
2
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50
J OHN M ICHAEL R ASSIAS
for all 0 ≤ k < j ≤
(9.26) Γq =
[q/2]
X
q
, 0 ≤ q ≤ p2 , p ∈ N. Let us denote
2
2k
2
k=0
2
1
1
Γ k+
Γ 2q − 2k +
2
2
X
q
q
k+j k+j
+2
(−1) 2
2k
2j
0≤k<j≤[q/2]
1
1
1
×Γ k+
Γ j+
Γ 2q − k − j +
.
2
2
2
q
2k
2
From (9.26) one gets
(9.27)
[q/2]
X
Ãqk + 2
k=0
1
B̃qkj = √ Γq .
π 2
0≤k<j≤[q/2]
X
Thus from (9.17) and (9.27) we find
(9.28)
for all 0 ≤ q ≤
(9.29)
|c0 |2 1
(m2q ) f (q) 2 = √ √ Γq ,
| |
c π 2
p
, p ∈ N, and c0 ∈ C, c > 0. Let us denote
2
[p/2]
X
∗
Γp = εp,q Γq ,
q=0
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]
4
X
2p
2p 1
∗ 2 |c0 |
Γ
,
(9.30)
Hp∗ 
εp,q (m2q ) f (q) 2  = Hp∗
p
2
| |
2π
c
q=0
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),
2
given as in the abstract, xm the mean of x for |f |2 , and ξm the mean of ξ for fˆ , then
E|f |2 ∗
|c0 |2 1 ∗
(9.31)
|Ep,f | = √ √ Γp = √ Γp ,
c π 2
π π
for any fixed but arbitrary p ∈ N. For instance, if c0 = 1 and c = 21 , 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
√
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
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
51
non-negative real function R : N → R, such that
Γ p+
R (p) =
Γ∗p
1
2
,
p ∈ N = {1, 2, 3, . . .}.
Then the extremum principle
R (p) ≥
(9.33)
1
,
2π
holds for any fixed but arbitrary p ∈ N. Equality holds for p = 1.
For instance, if p ∈ N9 = {1, 2, 3, . . ., 9}, then
1
429 1
≤ R (p) ≤
·
.
2π
23 2π
9.1. First nine cases of (9.33).
√
√
i) If p = 1, then q = 0. Thus Γ0 = Γ3 12 = π π, ε1,0 = −1, and Γ∗1 = |ε1,0 Γ0 | = π π.
√
1
But Γ 1 + 12 = 12 π. Hence R (1) = 2π
. Therefore the equality in (9.33) holds for
p = 1.
ii) If p = 2, then q = 0, 1.Thus
√
Γ0 = π π, ε2,0 = 2;
1
3 √
1
2
Γ1 = Γ
Γ 2+
= π π,
2
2
4
ε2,1 = −2,
and
1 √
Γ∗2 = |ε2,0 Γ0 + ε2,1 Γ1 | = π π.
2
√
1
But Γ 2 + 21 = 34 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
√
1
.
But Γ 3 + 21 = 15
π. Hence R (3) = 5 · 2π
8
Therefore the inequality in (9.33) holds for p = 3.
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52
J OHN M ICHAEL R ASSIAS
√
√
iv) If p = 4, then q = 0, 1, 2. Thus Γ0 = π π, ε4,0 = 24; Γ1 = 34 π π, 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
2j
2k
0≤k<j≤1
1
1
1
×Γ k +
Γ j+
Γ 4−k−j+
2
2
2
1
1
1
1
= Γ2
Γ 4+
+ 22 Γ2 1 +
Γ 2+
2
2
2
2
1
1
1
+ 2 (−1) 2Γ
Γ 1+
Γ 3+
2
2
2
57 √
105 + 12 − 60 √
=
π π = π π,
16
16
2k
2
2k
2
1
k+
2
because (k, j) ∈ {(0, 1)}, and
v)
vi)
vii)
viii)
ix)
Γ∗4 = |ε4,0 Γ0 + ε4,1 Γ1 + ε4,2 Γ2 |
39 √
3
57 √
= 24 + (−48) + 2
π π = π π.
4
16
8
√
But Γ 4 + 21 = 105
π. Hence R (4) = 35
· 1.
16
13 2π
Therefore the inequality in (9.33) holds for p√
= 4.
√
If p = 5, then q = 0, 1, 2. Thus Γ∗5 = 255
π
π. But Γ 5 + 12 = 945
π. Hence
16
32
63
1
R (5) = 17 · 2π . Therefore the inequality in (9.33) holds for p = 5.
√
√
If p = 6, then q = 0, 1, 2, 3. Thus Γ∗6 = 855
π π. But Γ 6 + 12 = 10395
π. Hence
32
64
1
231
R (6) = 19 · 2π . Therefore the inequality in (9.33) holds for p = 6.
√
1
135135 √
If p = 7, then q = 0, 1, 2, 3. Thus Γ∗7 = 7245
π
π.
But
Γ
7
+
=
π. Hence
64
2
128
1
R (7) = 429
·
.
Therefore
the
inequality
in
(9.33)
holds
for
p
=
7.
23
2π
√
√
If p = 8, then q = 0, 1, 2, 3, 4. Thus Γ∗8 = 192465
π π. But Γ 8 + 12 = 2027025
π.
128
256
495
1
Hence R (8) = 47 · 2π . Therefore the inequality in (9.33) holds for p = 8.
34459425 √
√
1
If p = 9, then q = 0, 1, 2, 3, 4. Thus Γ∗9 = 2344545
π
π.
But
Γ
9
+
= 512
π.
256
2
Hence R (9) =
In fact ,
12155
827
·
1
.
2π
ε9,0 = −9!,
ε9,3
Therefore the inequality in (9.33) holds for p = 9.
ε9,1
9 9! 6
=
,
6 6! 3
9!
=9· ,
2!
ε9,2
9 9!
=−
7 4!
7
2
,
ε9,4 = −81
and
Γ∗9 = |ε9,0 Γ0 + ε9,1 Γ1 + ε9,2 Γ2 + ε9,3 Γ3 + ε9,4 Γ4 | ,
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
53
√
√
√
where Γ0 = π π, Γ1 = 34 π π, and Γ2 = 57
π π from the above case iv). Besides
16
2 1
X
1
1
3
2
2k
2
Γ k+
Γ3 =
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
2 1
1
1
1
3
2
2
2
Γ 1+
=Γ
Γ 6+
+2
Γ 4+
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
because (k, j) ∈ {(0, 1)}, and
2 2
X
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.
Corollary 9.2. Assume that Γ is defined by (9.32). Consider the Gaussian f : R → C such that
2
f (x) = c0 e−cx , where c0 , c are fixed but arbitrary constants and c0 ∈ C, c > 0. Assume that
ˆ
xm is the mean of x for |f |2 . Consider
2f : R → C the Fourier transform of f , given asin the
2
abstract and ξm the mean of ξ 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]
Ep,f =
(9.34)
X
q=0
if |Ep,f | < ∞ holds for 0 ≤ q ≤
Then the extremum principle
(9.35)
Γ p + 12
Rf (p) =
|Ep,f |
p
2
and any fixed but arbitrary p ∈ N.
√
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
εp,q (m2q ) f (q) 2 ,
| |
=
π
2E|f |2
!
R∗ (p)
1
≥
|c0 |2
r
c
2
√
=
π
2E|f |2
!
,
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54
J OHN M ICHAEL R ASSIAS
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
1
c
429
1
c
, if p ∈ N9 .
≤ Rf (p) ≤
·
2
2
2
2
23 |c0 |
|c0 |
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
Rf (p) = Γ p +
|Ep,f |
2
,
|c |2
1
1
√ Γ∗p √0
=Γ p+
2
c
π 2
√
√
c
= π 2 2 R (p)
|c0 |
√
r
2c 1
c
1
≥π
=
.
2 ·
2
2
|c0 | 2π
|c0 |
Besides from (8.26) one gets
Z
Z
1
|c0 |2
|c0 |2 √
2
2
−2cx2
,
E1,f = − |f (x)| dx = − |c0 |
e
dx = − √
π = − √ 2Γ 1 +
2
2c
2c
R
R
p
or Rf (1) = |c1|2 2c , completing the proof of Corollary 9.2. We note that if c0 = 1, c = 12 , or
0
1 2
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.
9.2. First nine cases of (9.35).
i) If p = 1, then
|c0 |2 √
2
e−2cx dx = √
π.
2c
R
R
√
Thus from (8.26) we get E1,f = −E|f |2 . But Γ 1 + 12 = 12 π. Hence
r
c √ .
1
1
= π 2E|f |2 .
Rf (1) = Γ 1 +
|E1,f | =
2
|c0 |2 2
Z
E|f |2 =
2
2
Z
|f (x)| dx = |c0 |
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 E|f | = Γ0 = π π,
2 |E1,f | = π
|c0 |
|c0 |
√
respectively.
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
55
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
√
But Γ 2 + 21 = 34 π. Hence
r
1
1
c
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
√
3 √
2c 3
2c
2c
2
2
0
2
x |f (x)| dx = π
Γ1 = π
2 E|f | = π π,
2
2 (m2 )|f 0 |2 = π
4
|c0 | 4
|c0 | R
|c0 |
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
3
3
2
2
0
E3,f = −3 2E|f |2 − 3 x |f (x)| dx = −3 2E|f |2 − 3 · E|f |2 = E|f |2 .
4
4
R
√
But Γ 3 + 12 = 15
π. Hence
8
r
1
1
c
Rf (3) = Γ 3 +
|E3,f | = 5 ·
.
2
2
2
|c0 |
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
39
= 2 12 − 24 · +
E|f |2 = − E|f |2 < 0.
4 16
8
Hence
r
c
35
1
.
·
Rf (4) =
2
2
13 |c0 |
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 |
Hence
r
63
1
c
Rf (5) =
·
.
2
17 |c0 |
2
Therefore the inequality in (9.35) holds for p = 5.
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56
J OHN M ICHAEL R ASSIAS
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
3
57 2835
= 2 360 − 1080 · + 135 ·
−
E|f |2
4
16
64
855
E 2 < 0.
=−
32 |f |
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
0
0
4 00
6
0
+ 420 x f (x) dx − 7 x f (x) dx
R
R
57
2835
3
−7·
E|f |2
= −7 720 − 2520 · + 420 ·
4
16
64
7245
E 2 < 0.
=−
64 |f |
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
Z
0
2
2
2
0
E8,f = 2 20160E|f |2 − 80640 x |f (x)| dx + 16800 x4 f 0 (x) dx
R
R
Z
Z
2
2
6 000
8 (4)
−448 x f (x) dx + x f (x) dx
R
R
3
57
2835 273105
− 448 ·
+
E|f |2
= 2 20160 − 80640 · + 16800 ·
4
16
64
256
192465
=
E|f |2 .
128
Hence
495
1
·
Rf (8) =
47 |c0 |2
r
c
.
2
Therefore the inequality in (9.35) holds for p = 8.
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
57
ix) If p = 9, then from (8.46) one finds
Z
Z
0
2
2
2
0
E9,f = −9 40320E|f |2 − 181440 x |f (x)| dx + 45360 x4 f 0 (x) dx
R
R
Z
Z
2
2
8 (4)
6 000
−1680 x f (x) dx + 9 x f (x) dx
R
R
57
2835
273105
3
= 9 40320 − 181440 · + 45360 ·
− 1680 ·
+9·
E|f |2
4
16
64
256
2344545
=−
E|f |2 < 0.
256
Hence
r
12155
c
1
Rf (9) =
·
.
2
827 |c0 |
2
Therefore the inequality in (9.35) holds for p = 9.
We note that, from the Corollary 9.1,
Rf (p) = 2πR (p) = R∗ (p) ≥ 1
for any p ∈ N, if
−2x2
f (x) = e
because |Ep,f | =
1 ∗
Γ,
2π p
or E|f |2
√ π
=
,
2
from (9.31).
Corollary 9.3. Assume that the Euler gamma function Γ is defined by (9.32). Consider the
2
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
ˆ
Consider the Fourier
of this paper, and ξm
2 transform f : R → C of f , given as in the abstract
2
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
[p/2]
Ep,f =
X
q=0
if |Ep,f | < ∞ for 0 ≤ q ≤
εp,q (m2q ) f (q) 2 ,
| |
p
, and any fixed but arbitrary p ∈ N. If
2
2cx20
d2p 2cx20
e
dx2p
0
(> 0) denotes
the 2pth order derivative of e
with respect to x0 , then the extremum principle
12
2p
√
4
1
1
π
|c0 |2 −cx20
d
2
2cx
e 0
(9.36)
|Ep,f | ≤ 3p−1 · Γ 2 p +
· p+1 · e
·
,
2
dx2p
2 2
c 2
0
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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58
J OHN M ICHAEL R ASSIAS
Proof. At first, we claim that the general integral formula
√
Z
1
d2p
π
2
2
2p −2c(x−x0 )2
(9.37)
x e
dx = 4p+ 1 · 2p+ 1 · e−2cx0 · 2p e2cx0 , c > 0, x0 ∈ R,
dx0
2 2 c 2
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
2cx2
d d2 2cx20
2cx20
2
0 = 4c.
e
=
4cx
e
=
4c
1
+
4cx
0
0 e
dx20
dx0
Thus (9.37) yields
Z
2 −2cx2
xe
R
This equals to
Γ 1+ 12
1
(2c)1+ 2
(
)
1
dx =
(4c)2
r
π
1
· 1 · 4c =
2c
4c
r
π
.
2c
because
√
3
1
1
π
Γ
= Γ
=
2
2
2
2
implying (9.1). A direct proof for this goes, as follows:
Z
Z
1
2
2 −2cx2
xe
dx =
xd e−2cx
−4c R
R
Z
1
−2cx2
−2cx2
=
xe
|R − e
dx
−4c
R
Z
1
2
=
e−2cx dx
4c R
Z
√ √
2
1
e−( 2cx) d
2cx
= √
4c 2c R
1 √
π,
= √
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
2
= |c0 |
xe−2c(x−x0 ) dx
R
|c0 |2
=−
4c
Z
−2c(x−x0 )2
d e
2
Z
+ x0 |c0 |
R
2
e−2c(x−x0 ) dx,
R
or
r
(9.39)
xm = x0 E|f |2 =
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
π |c0 |2
√ x0 ,
2 c
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
59
because
Z
|f (x)|2 dx
R
Z
2
2
= |c0 |
e−2c(x−x0 ) dx
E|f |2 =
R
2
|c0 |
=√
2c
Z
√
e−(
2
2c(x−x0 ))
d
√
2c (x − x0 ) ,
R
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
xm = 0.
(9.41)
2
In this case the mathematical expectation xm is the√mean
of x for |f | only if x0 = 0.
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
(9.42)
2p
d2p
p d
(·)
=
(−1)
(·)
dx2p
d (ix0 )2p
0
then he proves
d2p x20
e , p ∈ N0 .
dx2p
0
Therefore from (9.38) with n = 2p, and (9.43) one gets that the integral formula
√
Z
π
d2p 2
2
2p −(x−x0 )2
(9.44)
x e
dx = 2p · e−x0 · 2p ex0 ,
2
dx0
R
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 )
s
e
ds
x e
dx =
1 ·
R
R
(2c)p+ 2
√
√
2
d2p
π −(√2cx0 )2
1
( 2cx0 )
·
·
e
e
=
√
1 ·
2p
2p
d 2cx0
(2c)p+ 2 2
√
d2p
π 1
2
2cx2
(9.45)
= 3p+ 1 p+ 1 · e−2cx0 · √
2p e 0 , c > 0, x0 ∈ R,
2 2c 2
d 2cx0
(9.43)
2
H2p (ix0 ) = (−1)p e−x0
holds for all p ∈ N0 .
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J OHN M ICHAEL R ASSIAS
However,
dx0
d
d
1
d
√
(·) =
=
(·) √
(·) ,
1 ·
dx0
d 2cx0
d 2cx0
(2c) 2 dx0
and
(9.46)
1 d2p
d2p
(·)
=
(·) ,
√
2p
(2c)p dx2p
d 2cx0
0
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
2
In fact, differentiating the Gaussian function f : R → C of the form f (x) = c0 e−c(x−x0 ) with
respect to x, one gets
f 0 (x) = −2c (x − x0 ) f (x) = −2cxf (x) + 2cx0 f (x) .
Thus the Fourier transform of f 0 is
∧
F f 0 (ξ) = F [f 0 (x)] (ξ) = [f 0 (x)] (ξ) = [−2cxf (x)]∧ (ξ) + [2cx0 f (x)]∧ (ξ) ,
or
−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
2iπξ fˆ (ξ) =
f (ξ) + 2cx0 fˆ (ξ) ,
iπ
or
−2π 2 ξ fˆ (ξ) = cfˆ0 (ξ) + 2iπcx0 fˆ (ξ) ,
(9.48)
2iπξ fˆ (ξ) =
or
0
2π
ˆ
(9.49)
f (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
(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
0
− πc ξ 2 −i2πx0 ξ
ˆ
(9.51)
f (ξ) = e
K (ξ) + K (ξ) −
ξ − i2πx0 .
c
From (9.48), (9.49), (9.50) and (9.51) we find 0 = K 0 (ξ) e−
(9.52)
π2 2
ξ −i2πx0 ξ
c
, or K 0 (ξ) = 0, or
K (ξ) = K,
which is a constant. But from (9.50) and (9.52) one gets
(9.53)
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
fˆ (0) = K (0) = K.
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
61
Besides from the definition of the Fourier transform we get
Z
fˆ (0) =
e−2iπ·0·x f (x) dx
ZR
=
f (x) dx
R
Z
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
2
of (9.47) is by employing the formula (9.2) for the special Gaussian φ (x) = c0 e−cx , such that
r
π − π2 ξ 2
φ̂ (ξ) = c0
e c , c0 ∈ C, c > 0.
c
In fact, f (x) = φ (x − x0 ), or
Z
Z
∧
−2iπξx
ˆ
f (ξ) = [φ (x − x0 )] (ξ) =
e
φ (x − x0 ) dx =
e−2iπξ(x+x0 ) φ (x) dx
R
R
(with x + x0 on x)
r
Z
π − π2 ξ 2
−2iπξx0
−2iπξx
−2iπξx0
−2iπx
ξ
0
e c ,
=e
e
φ (x) dx = e
φ̂ (ξ) , or fˆ (ξ) = e
c0
c
R
establishing (9.47).
2
Therefore from (8.36) – (8.37) with xm = 0, from (9.41), and the mean of ξ for fˆ of the
form
Z Z
π2 2
ˆ 2
2 π
ξm =
ξ f (ξ) dξ = |c0 |
ξ · e−2 c ξ dξ = 0,
c R
R
as well as from (9.1), (9.37) and (9.47), one finds that the left-hand side of the inequality (8.40)
of Corollary 8.2 is
Z
Z
2 2
2p
2p ˆ
(m2p )|f |2 (m2p ) fˆ 2 =
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 |
x e
dx ·
ξ e
dξ
where c =
c
c
R
R
!
√
Γ p + 21
d2p 2cx20
1
π
4 π
−2cx20
·e
· 2p e
·
= |c0 |
1
c 24p+ 12 c2p+ 12
dx0
(2c∗ )p+ 2
√
π
1
|c0 |4 −2cx20 d2p 2cx20
∗ 2p
Γ
p
+
·
·e
e ,
= Hp
23p−1
2
cp+1
dx2p
0
√ with Hp∗ = 1 2π p 2 holds for all p ∈ N, c0 ∈ C, c > 0, and x0 ∈ R.
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62
J OHN M ICHAEL R ASSIAS
Finally from the right-hand side of the inequality (8.40) of Corollary 8.2 with
[p/2]
Ep,f =
X
Z
εp,q
2
x f (q) dx
2q
R
q=0
such that |Ep,f | < ∞ and
(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). Corollary 9.4. Assume that the Euler gamma function Γ is defined by (9.32). Consider the
2
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 .
ˆ
Consider
f , given as in the abstract, and ξm the mean of ξ
2 the Fourier transform f : R → C of
(q) 2
ˆ
th
for f . Consider the 2q moment of x for f by
Z
(µ2q ) f (q) 2 =
| |
2
(x − xm )2q f (q) (x) dx,
R
the constants εp,q as in (8.38), and
[p/2]
Ep,f =
X
q=0
εp,q (µ2q ) f (q) 2 ,
| |
p
, and any fixed but arbitrary p ∈ N. If
2
!
r
2
π
|c
|
0
√
E|f∗ |2 = 1 − E|f |2 = 1 −
2 c
if |Ep,f | < ∞ holds for 0 ≤ q ≤
and x∗0 = x0 E|f∗ |2 , then the extremum principle
√
4
|Ep,f | ≤
(9.56)
2
π
3p−1
2
·Γ
1
2
1
p+
2
2p
21
2
|c0 |2 ∗ −p −cx∗2
d
2cx∗0
0
· p+1 · E|f |2
·e
·
e
,
dx2p
c 2
0
holds for any fixed but arbitrary p ∈ N.
Equality holds for p = 1; x0 = 0, or for p = 1; E|f |2 = 1.
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 Z
2 2p
2
2p ˆ
=
x − x0 E|f |2
|f (x)| dx ·
ξ f (ξ) dξ
where xm = x0 E|f |2
R
4
|c0 |
=
π
c
R
Z x − x0 E|f |2
2p
R
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
−2c(x−x0 )2
e
Z
π2
2p −2c∗ ξ 2
∗
dx ·
ξ e
dξ
where c =
.
c
R
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
63
By placing x + x0 E|f |2 on x and letting x∗0 = x0 1 − E|f |2 = x0 E|f∗ |2 we have
!
Z
1
2
Γ
p
+
|c0 |4
∗
2
x2p e−2c(x−x0 ) dx
(µ2p )|f |2 (m2p ) fˆ 2 = π
p+ 12
| |
∗
c
R
(2c )
!
√
1
2p
2
2
Γ
p
+
π
1
d
|c0 |4
∗
∗
2
· e−2cx0 · ∗2p e2cx0
.
=π
1
4p+ 12 2p+ 12
p+
2
c
dx0
2
c
(2π /c) 2
However,
1
d2p
d2p
(·)
=
2p 2p (·) ,
2p
dx∗0
dx0
∗
E|f |2
and
Hp∗
2p
=
1
22(p+1) π 2p
.
Therefore
(9.57)
(m2p )|f |2 (m2p ) fˆ 2
| |
√
π
1
|c0 |4 ∗ −2p −2cx∗02 d2p 2cx∗02
∗ 2p
= Hp
Γ p+
· p+1 E|f |2
·e
· 2p e
23p−1
2
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
∗
2
d2 2cx∗02
d
∗ 2cx∗0 dx0
e
=
4cx0 e
dx20
dx0
dx0
2
d ∗ 2cx∗02 dx∗0
= 4c ∗ x0 e
dx0
dx0
dx∗
2
2
∗2
= 4cE|f∗ |2 1 + 4cx∗0 e2cx0 , where 0 = E|f∗ |2 .
dx0
∗2
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
s √
4
π
3 |c0 |2 ∗ −1 q
2
∗
R.H.S. (for p = 1; x0 = 0 or x0 = 0) =
Γ
E|f |2
(9.58)
4cE|f∗ |2
2
2
c
r
π |c0 |2
√ = 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
|E1,f | = E|f |2 .
(9.59)
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.
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J OHN M ICHAEL R ASSIAS
Besides we note from (9.56) at x0 6= 0 one gets that
s √
i 12
4
3 |c0 |2 ∗ −1 h
π
∗2
∗2
(9.60) R.H.S. (for p = 1; x0 6= 0) =
Γ
E|f |2
4cE|f |2 1 + 4cx0
2
2
c
2 12
= E|f |2 1 + 4c 1 − E|f |2 x20 .
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
2 21
(9.61)
|E1,f | = E|f |2 ≤ 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.
(ii) If p = 2, then one gets
2 2
d4 2cx∗02
∗2 d
∗2
2cx∗0
(9.62)
e
= 4cE|f |2 2
1 + 4cx0 e
dx40
dx0
2
d ∗
2 ∗3
∗3
2cx∗0
= 16c E|f |2
3x0 + 4cx0 e
dx0
∗2
2 ∗4
∗2
2 ∗4
= 16c E|f |2 3 + 24cx0 + 16c x0 e2cx0 .
Therefore at x0 = 0 (or x∗0 = 0), we have
d4 2cx∗02
4
e
= 48c2 E|f∗ |2 .
4
dx0
(9.63)
Thus
s √
4
π
5 |c0 |2 ∗ −2 q 2 ∗4
E|f |2
48c E|f |2 ,
R.H.S. (for p = 2; x0 = 0 (or x∗0 = 0)) = 5 Γ
2 c 32
22
or
r
3
3 π |c0 |2
√ = E|f |2 , for p = 2; x0 = 0.
(9.64)
R.H.S. =
2 2 c
2
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
|f (x)| − x |f (x)| dx = 2 E|f |2 − E|f |2 ,
E2,f = 2
4
R
or
1
(9.65)
|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
1
3
(9.66)
|E2,f | = E|f |2 < E|f |2 .
2
2
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
65
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
2
2
= E|f |2 3 + 24c 1 − E|f |2 x0 + 16c 1 − E|f |2 x40 .
2
We note that
Z
2
2
0
R
Z 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
+ 1 + 4E|f |2 + E|f2 |2 x20 x2 |f (x)|2 dx
R
Z
Z
2
2
2
3
2
4
− 2 E|f |2 + E|f |2 x0 x |f (x)| dx +E|f |2 x0 |f (x)| dx .
(x − xm ) |f (x)| dx = 4c
2
x − x0 E|f |2
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 ,
ZR
1 + 4cx20
x2 |f (x)|2 dx = (m2 )|f |2 =
E|f |2 ,
4c
R
Z
3x0 + 4cx30
x3 |f (x)|2 dx = (m3 )|f |2 =
E|f |2 ,
4c
R
and
Z
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
2
2 4
2
2
= E|f | 3 + 24cx0 + 16c x0 − 8c 1 + E|f |
3x20 + 4cx40
4
+ 4c 1 + 4E|f |2 + E|f2 |2 x20 + 4cx40
i
2
2
4
2 2
4
−32c E|f |2 + E|f |2 x0 +16c E|f |2 x0
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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66
J OHN M ICHAEL R ASSIAS
or
Z
2
(x − xm )2 |f 0 (x)| dx =
R
2
3 + 4c 1 − E|f |2 x20
4
E|f |2 ,
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
(1)(0)
E2,f = (−1)
0
0
2−0
0!
R
2
0
× (x − xm ) (x − xm )2·0 f (0) (x) dx
Z
2
2! 0
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
2
0
=2
|f (x)| dx − (x − xm ) |f (x)| dx
R


2
2
2
3
+
4c
1
−
E
x
0
|f |


= 2 E|f |2 −
E|f |2  ,
4
or
E2,f
(9.68)
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 p = 2 such that
2 1
(9.69)
|E2,f | = E|f |2 1 − 4c 1 − E|f |2 x20 2
2
4 12
3
2
2
< E|f |2 3 + 24c 1 − E|f |2 x0 + 16c 1 − E|f |2 x40 ,
2
because the condition
2 2
2 2
4 4c 1 − E|f |2 x0 + 28 4c 1 − E|f |2 x20 + 13 > 0,
or
(9.70)
4
2
64c2 1 − E|f |2 x40 + 112c 1 − E|f |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). Consider the general
2
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
and c0 ∈ C, c1 > 0, and c2 , c3 ∈ R. Assume that the mathematical
constants
c2
expectation E x − 2c1 of x − 2cc21 for |f |2 equals to
Z c2
xm =
x−
|f (x)|2 dx = 0.
2c1
R
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
67
ˆ
Consider
2 the Fourier transform f : R → C of f , given as in the abstract, and ξm the mean of ξ
2
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
[p/2]
Ep,f =
X
q=0
p ∈ N.
εp,q (m2q ) f (q) 2 ,
| |
2
If ε0 = c0 e(c2 +4c1 c3 )/4c1 ∈ C and t0 = 2cc21 ∈ R, then the extremum principle
2p
1
√
4
1
d 2c1 t20 2
π
1
|ε0 |2 −c1 t20
·
,
· p+1 · e
(9.71)
|Ep,f | ≤ 3p−1 · Γ 2 p +
2p e
2
2
dt
2 2
0
c1
holds for any fixed but arbitrary p ∈ N.
Equality holds for p = 1; t0 = 0 (or c2 = 0).
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)
2
f (x) = d0 g (x) = ε0 e−c1 (x−t0 ) ,
ε0 ∈ C.
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|
or
(9.73)
xm = 0.
In this case the mathematical expectation xm is the mean of x for |f |2 if t0 = 0 or c2 = 0.
We note that from (9.40) and (9.72) one establishes
r
r
1 c3
π |c0 |2
π |c0 |2 c22 +4c
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
which leads to (9.74).
Similarly from (9.39) we get the mean of f for |f |2 of the form
r
r
Z
π |ε0 |2
1 π |c0 |2 c2 (c22 +4c1 c3 )/2c1
2
(9.75)
x |f (x)| dx = t0 E|f |2 =
.
√ t0 =
√ e
2 c1
2 2 c1 c1
R
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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68
J OHN M ICHAEL R ASSIAS
Also from (9.47) one finds the Fourier transform fˆ of f of the form
r
r
π2 2
π
π (c22 +4c1 c3 )/4c1 −(π2 ξ2 +iπc2 ξ)/c1
ξ
−i2πt
ξ
−
0
e c1
= c0
e
e
,
(9.76)
fˆ (ξ) = ε0
c1
c1
c0 ∈ C, c1 > 0, and c2 , c3 ∈ R.
2
Finally we find the mean of ξ for fˆ , as follows:
Z
Z 2
ˆ 2
−2 π ξ 2
2 (c22 +4c1 c3 )/2c1 π
(9.77)
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 general Gaussian
2
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, and c2 , c3 ∈ R. Assume that xm is the mean (or the
ˆ
mathematical expectation E (x)) of x for |f |2 . Consider
2 the Fourier transform f : R → C of f ,
given as 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
[p/2]
Ep,f =
X
q=0
holds for 0 ≤ q ≤
if |Ep,f | < ∞
εp,q (µ2q ) f (q) 2 ,
| |
p
, and any fixed but arbitrary p ∈ N. If
2
2
ε0 = c0 e(c2 +4c1 c3 )/4c1 ∈ C,
and
t0 =
c2
∈ R,
2c1
E|f∗ |2 = 1 − E|f |2
r
=1−
π |ε0 |2
√
2 c1
!
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
1
p+
2
2p
12
d 2c1 t∗2
|ε0 |2 ∗ −p −c1 t∗0 2
0
·e
·
,
· p+1 · E|f |2
2p e
2
dt
0
c1
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
2
From (9.74) – (9.75) one gets that xm = t0 E|f |2 , where E|f |2 = π2 |ε√0c|1 . 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,
2
because from (9.72) we have f (x) = ε0 e−c1 (x−t0 ) , and xm is the mean of x for |f |2 .
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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O N THE H EISENBERG -PAULI -W EYL INEQUALITY
69
R EFERENCES
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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
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[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
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[8] C. GASQUET
York, 1998).
AND
P. WITOMSKI, Fourier Analysis and Applications, (Springer-Verlag, New
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27 (1889); Scientific Papers, Cambridge University Press, Cambridge, England, 1902, and Dover,
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70
J OHN M ICHAEL R ASSIAS
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Boston, Mass., 1982.
Table 9.1: 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
22
23
24
25
26
27
28
29
30
31
32
33
J. Inequal. Pure and Appl. Math., 5(1) Art. 4, 2004
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
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)
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
3158.42
4584.05
2655.83
3739.60
12647.30
18286.41
10611.98
14951.80
50630.40
72997.06
42413.73
59787.71
http://jipam.vu.edu.au/