Journal of Inequalities in Pure and
Applied Mathematics
ON THE REFINED HEISENBERG-WEYL TYPE INEQUALITY
JOHN MICHAEL RASSIAS
National and Capodistrian University of Athens
Pedagogical Department EE
Section of Mathematics,4, Agamemnonos Str.
Aghia Paraskevi
Athens 15342, Greece
volume 6, issue 2, article 45,
2005.
Received 11 January, 2005;
accepted 17 March, 2005.
Communicated by: S. Saitoh
EMail: jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/∼jrassias/
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
013-05
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Abstract
The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f : R → C is a complex valued function of a
random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the
product of the second moment of the random real x for |f|2 and the second
2
.
moment of the random real ξ for fb is at least ER,|f|2 4π, where fb is the
R
R
Fourier transform of f, fb(ξ) = R e−2iπξx f (x) dx and f (x) = R e2iπξx fˆ (ξ) dξ,
R
and ER,|f|2 = R |f (x)|2 dx. This uncertainty relation is well-known in classical
quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined
form of the generalized Heisenberg-Weyl type inequality is established.
On the Refined
Heisenberg-Weyl Type
Inequality
John Michael Rassias
2000 Mathematics Subject Classification: 26, 33, 42, 60, 62.
Key words: Heisenberg-Weyl Type Inequality, Uncertainty Principle, Gram determinant.
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Second Moment Heisenberg-Weyl Inequality . . . . . .
1.2
Fourth Moment Heisenberg-Weyl Inequality . . . . . . .
1.3
Second Moment Heisenberg-Weyl Type Inequality . .
1.4
Fourth Moment Heisenberg-Weyl Type Inequality . .
2
Refined Heisenberg-Weyl Type Inequality . . . . . . . . . . . . . . . .
3
Applied Refined Heisenberg-Weyl Type Inequality . . . . . . . .
3.1
Refined 2nd Mom. Heisenberg-Weyl Type Inequality
3.2
Refined 4th Mom. Heisenberg-Weyl Type Inequality
References
3
3
5
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1.
Introduction
The serious question of certainty in science was high-lighted by Heisenberg, in
1927, via his “uncertainty principle” [2]. He demonstrated, for instance, the
impossibility of specifying simultaneously the position and the speed (or the
momentum) of an electron within an atom. In 1933, according to Wiener [7] “a
pair of transforms cannot both be very small.” This uncertainty principle was
stated in 1925 by Wiener, according to Wiener’s autobiography [8, p. 105–
107], in a lecture in Göttingen. The following result of the Heisenberg-Weyl
Inequality is credited to Pauli according to Weyl [6, p. 77, p. 393–394]. In
2
1928, according to
2 Pauli [6] “ the less the uncertainty in |f | , the greater the
uncertainty in fb , and conversely.” This result does not actually appear in
Heisenberg’s seminal paper [2] (in 1927).
In 1998, Burke Hubbard [1] wrote a remarkable book on wavelets. According to her, most people first learn the Heisenberg uncertainty principle in
connection with quantum mechanics, but it is also a central statement of information processing. The following second order moment Heisenberg-Weyl
inequality provides a precise quantitative formulation of the above-mentioned
uncertainty principle.
1.1.
Second Moment Heisenberg-Weyl Inequality ([1], [4], [5])
For any f ∈ L2 (R), f : R → C, such that
Z
2
kf k2,R =
|f (x)|2 dx = ER,|f |2 ,
R
On the Refined
Heisenberg-Weyl Type
Inequality
John Michael Rassias
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any fixed but arbitrary constants xm , ξm ∈ R, and for the second order moments
Z
2
(µ2 )R,|f |2 = σR,|f |2 =
(x − xm )2 |f (x)|2 dx
R
and
(µ2 )R, fb 2 =
| |
σ2 b 2
R,|f |
Z
=
R
2
(ξ − ξm )2 fb(ξ) dξ,
the second order moment Heisenberg-Weyl inequality
(H1 )
2
σR,|f
· σ2
|2
2
R,|fb|
≥
kf k42,R
,
16π 2
holds. Equality holds in (H1 ) if and only if the generalized Gaussians
f (x) = co exp (2πixξm ) exp −c (x − xm )2
hold for some constants co ∈ C and c > 0.
The Heisenberg-Weyl inequality in spectral analysis says that the product of
the effective duration ∆x and the effective bandwidth ∆ξ of a signal cannot
be
.
2
less than the value 1/4π, where ∆x2 = σR,|f
ER,|f |2 and ∆ξ 2 = σ 2 b 2 ER,|f |2
|2
R,|f |
with f : R → C, fb : R → C defined as in (H1 ), and
Z
Z b 2
2
(PPR)
ER,|f |2 =
|f (x)| dx =
f (ξ) dξ = ER,|fb|2
R
R
according to the Plancherel-Parseval-Rayleigh identity.
On the Refined
Heisenberg-Weyl Type
Inequality
John Michael Rassias
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1.2.
Fourth Moment Heisenberg-Weyl Inequality
([4, pp. 26–27])
For any f ∈ L2 (R), f : R → C, such that
Z
2
kf k2,R =
|f (x)|2 dx = ER,|f |2 ,
R
any fixed but arbitrary constants xm , ξm ∈ R, and for the fourth order moments
Z
(µ4 )R,|f |2 =
(x − xm )4 |f (x)|2 dx
On the Refined
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R
John Michael Rassias
and
Z
(µ4 )R, fb 2 =
| |
R
2
4b
(ξ − ξm ) f (ξ) dξ,
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the fourth order moment Heisenberg-Weyl inequality
Contents
1
(µ4 )R,|f |2 · (µ4 )R, fb 2 ≥
E2 ,
| |
64π 4 2,R,f
(H2 )
JJ
J
holds, where
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E2,R,f = 2
Z h
2
2
xδ |f (x)|2 − x2δ |f 0 (x)|
1 − 4π 2 ξm
2
Close
R
−
4πξm x2δ
Im
i
f (x)f 0 (x)
dx,
with xδ = x − xm , ξδ = ξ − ξm , Im(·) is the imaginary part of (·), and
|E2,R,f | < ∞.
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The “inequality” (H2 ) holds, unless f (x) = 0.
We note that if the ordinary differential equation of second order
(ODE)
fα00 (x) = −2c2 x2δ fα (x)
holds, with α = −2πξm i, fα (x) = eαx f (x), and a constant c2 = 12 k22 > 0,
k2 ∈ R and k2 6= 0, then “equality” in (H2 ) seems to occur. However, the
solution of this differential equation (ODE), given by the function
p
1
1
2πixξm
2
2
f (x) = |xδ |e
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
2
2
in terms of the Bessel functions J±1/4 of the first kind of orders ±1/4, leads to
a contradiction, because this f ∈
/ L2 (R). Furthermore, a limiting argument is
required for this problem. For the proof of this inequality see [4]. It is open
to investigate cases, where the integrand on the right-hand side of the integral
of E2,R,f will be nonnegative. For instance, for xm = ξm = 0, this integrand
is:= |f (x)|2 − x2 |f 0 (x)|2 (≥ 0).
In 2004, we [4] generalized the Heisenberg-Pauli-Weyl inequality in R =
(−∞, ∞). In this paper, a refined form of this generalized Heisenberg-Weyl
type inequality is established in I = [0, ∞). Afterwards, an open problem is
proposed on some pertinent extremum principle. However, the above-mentioned
Fourier transform is considered in R, while our results in this paper are restricted to I = [0, ∞). Futhermore, the corresponding inequality is investigated in R, as well. Our second moment Heisenberg-Weyl type inequality and
the fourth moment Heisenberg-Weyl type inequality are of the following forms
(Ri ), (i = 1, 2).
On the Refined
Heisenberg-Weyl Type
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John Michael Rassias
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1.3.
Second Moment Heisenberg-Weyl Type Inequality ([4])
R
For any f ∈ L2 (I), I = [0, ∞), f : I → C, such that kf k22,I = I |f (x)|2 dx =
EI,|f |2 , any fixed but arbitrary constant xm ∈ R, and for the second order
moment
Z
2
(µ2 )I,|f |2 = σI,|f |2 = (x − xm )2 |f (x)|2 dx,
I
the second order moment Heisenberg-Weyl type inequality
Z
2
1
1 2
2
0 2
(R1 )
(µ2 )I,|f |2 · kf k2,I ≥ E1,I,f =
− |f (x)| dx ,
4
4
I
holds, where |E 1,I,f | < ∞. Equality
holds in (R1 ) if and only if the Gaussians
f (x) = co exp −c (x − xm )2 hold for some constants co ∈ C and c > 0.
We note that this inequality (R1 ) still holds if we replace the interval of
integration I with R, without any other change.
1.4.
Fourth Moment Heisenberg-Weyl Type Inequality ([4])
For any f ∈ L2 (I), I = [0, ∞), f : I → C, such that kf k22,I = I |f (x)|2 dx =
EI,|f |2 , any fixed but arbitrary constant xm ∈ R, and for the fourth order moment
Z
(µ4 )I,|f |2 = (x − xm )4 |f (x)|2 dx,
R
I
the fourth order moment Heisenberg – Weyl type inequality
Z h
i 2
1 2
2
2
00 2
2
0
(R2 ) (µ4 )I,|f |2 · kf k2,I ≥ E2,I,f =
|f (x)| dx − xδ |f (x)| dx
4
I
On the Refined
Heisenberg-Weyl Type
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John Michael Rassias
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holds, where xδ = x − xm , and |E2,I,f | < ∞.
The “inequality” (R2 ) holds, unless f (x) = 0.
We note that this inequality (R2 ) still holds if we replace the interval of
integration I with R, without any other change except that one on the following
condition (2.1), where x → ∞ has to be substituted with |x| → ∞.
We omit the proofs of the inequalities (Ri ) (i = 1, 2) as special cases of the
corresponding proof of the following general Theorem 2.1 (with A = 0) of this
paper. Furthermore, we state our following four pertinent propositions. Their
proofs are identical or analogous to the proofs of the corresponding propositions
of [4].
Proposition 1.1 (Pascal type combinatorial identity, [4]). If 0 ≤ k2 is the
greatest integer ≤ k2 , then
k−i
k−1 k−i
k+1
k−i+1
k
+
=
,
(C)
k−i
i
k−i i−1
k−i+1
i
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, and 0 ≤ i ≤ k2 for
k
i ∈ N0 = {0, 1, 2, . . .} such that −1
= 0.
Proposition 1.2 (Generalized differential identity, [4]). If f : I → C is a
complex valued function of a real variable x, I = [0, ∞), 0 ≤ k2 is the
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
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John Michael Rassias
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k
(−1)
k−i
i
k−i
i
2
dk−2i (i)
,
f
(x)
dxk−2i
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holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤
i ∈ N0 = {0, 1, 2, . . .}.
k
2
for
We note that the proof of (*) requires the application of the new identity (C).
Furthermore, we note that the above differential identity (*) still holds if we
replace the interval of integration I with R, without any other change.
Proposition 1.3 (P th -derivative of product, [4]). If fi : I → C (i = 1, 2) are
two complex valued functions of a real variable x, then the pth -derivative of the
(m)
(p−m)
product f1 f2 is given, in terms of the lower derivatives f1 , f2
by
(p)
(f1 f2 )
(1.1)
=
p X
p
m=0
m
(m) (p−m)
f1 f2
On the Refined
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John Michael Rassias
for any fixed but arbitrary p ∈ N0 .
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Proposition 1.4 (Generalized integral identity, [4]). If f : I → C is a complex
valued function of a real variable x, I = [0, ∞), and h : I → R is a real valued
function of x, as well as, w, wp : I → R are two real valued functions of x,
p
such that wp (x) = (x − xm
p) w(x) for any fixed but arbitrary constant xm ∈ R
and v = p − 2q, 0 ≤ q ≤ 2 , then
i)
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Z
(1.2)
wp (x) h(v) (x) dx
=
v−1
X
r=0
r
(−1)
wp(r)
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(v−r−1)
(x) h
v
(x) + (−1)
Z
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wp(v)
(x) h (x) dx
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holds for any fixed but arbitrary p ∈ N0 and v ∈ N, and all r : r =
0, 1, 2, . . ., v − 1, as well as the integral identity
ii)
Z
(v)
wp (x) h
v
Z
(x) dx = (−1)
I
wp(v) (x) h (x) dx
I
holds if the limiting condition
iii)
v−1
X
r=0
(−1)r lim wp(r) (x) h(v−r−1) (x) = 0,
x→∞
holds, and if all of these integrals exist.
We note that the proof of (1.2) requires the application of the differential
identity (1.1). Furthermore, we note that the above integral identity ii) still holds
if we replace the interval of integration I with R, without any other change
except that on the above limiting condition iii), where x → ∞ has to be
substituted with |x| → ∞.
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John Michael Rassias
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2.
Refined Heisenberg-Weyl Type Inequality
We assume that f : I → C is a complex valued function of a real variable x,
and w : I → R a real valued weight function of x, as well as xm any fixed but
arbitrary real constant. Also we denote
Z
(µ2p )w,I,|f |2 = w2 (x) (x − xm )2p |f (x)|2 dx
I
the 2pth weighted moment of x for |f |2 with weight function w : I → R.
Besides we denote
p
p−q
q
Cq = (−1)
,
p−q
q
if 0 ≤ q ≤ p2 (= the greatest integer ≤ p2 ),
Z
2
p−2q
Iql = (−1)
wp(p−2q) (x) f (l) (x) dx,
I
if 0 ≤ l ≤ q ≤ p2 , and wp = (x
xm )p w. We assume that all these integrals
P−
exist. Finally we denote Dq = ql=0 Iql , if |Dq | < ∞ holds for 0 ≤ q ≤ p2 ,
and
[p/2]
X
Ep,I,f =
Cq Dq ,
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John Michael Rassias
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q=0
if |Ep,I,f | < ∞ holds for p ∈ N. In addition, we assume the condition:
p−2q−1
(2.1)
X
r=0
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2 (p−2q−r−1)
r
(r)
(l)
(−1) lim wp (x) f (x)
= 0,
x→∞
J. Ineq. Pure and Appl. Math. 6(2) Art. 45, 2005
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for 0 ≤ l ≤ q ≤
p
. Furthermore,
2
∗
|Ep,I,f
|=
(2.2)
q
2
Ep,I,f
+ 4A2 ,
where A =R kuk x0 − kvk y0 , with L2 −norm k·k2 =
(|u| , |v|) = I |u| |v|, and
u = w(x)xpδ f (x),
Z
|v(x)h(x)|dx,
x0 =
R
I
|·|2 , inner product
v = f (p) (x);
Z
y0 = |u(x)h(x)|dx,
I
I
as well as
1
h(x) = √
σ
r
4
John Michael Rassias
2 − 1 ( x−µ )2
e 4 σ ,
π
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or
(HI )
h(x) =
√
1
2√
4
nπ
s
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Contents
Γ( n+1
)
2
n
Γ( 2 )
1
·
1+
x2
n
,
n+1
4
where µ is the mean, σ the standard deviation, and n ∈ N, and
Z
2
kh(x)k = |h(x)|2 dx = 1.
I
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2
Theorem 2.1. If (2.1) holds and f ∈ L (R), then
q
q
1 q
p ∗
p
∗
2p
(p)
,
(Rp )
(µ2p )w,I,|f |2 kf k2,I ≥ √
E
p,I,f
p
2
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holds for any fixed but arbitrary p ∈ N.
Equality holds in (Rp∗ ) iff v(x) = −2cp u(x) holds for constants cp > 0, and
any fixed but arbitrary p ∈ N; cp = kp2 /2 > 0, kp ∈ R and kp 6= 0, p ∈ N,
and A = 0, or h(x) = c1p u(x) + c2p v(x) and x0 = 0, or y0 = 0, where cip
(i = 1, 2) are constants and A2 > 0.
We note that this inequality (Rp∗ ) still holds if we replace the interval of
integration I with R, without any other change except that one on the above
condition (2.1), where x → ∞ has to be substituted with |x| → ∞, and the
choice of h from (HI ) must be replaced with
h(x) = √
4
1 x−µ 2
1
√ e− 4 ( σ ) ,
2π σ
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John Michael Rassias
or
1
h(x) = √
4
nπ
(HR )
s
Γ n+1
1
2
·
n+1 ,
n
2
Γ 2
1 + xn 4
where µ is the mean, σ the standard deviation, and n ∈ N.
Proof. In fact, one gets
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2
· f (p) 2,I − A2
= (µ2p )w,I,|f |2
Z
Z
(p) 2
2p
2
2
f (x) dx − A2
=
w (x) (x − xm ) |f (x)| dx ·
(2.4)
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(2.3) Mp∗ = Mp − A2
I
2
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= kuk kvk2 − A2
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with u = w(x)xpδ f( x), v = f (p) (x), where xδ = x − xm .
From (2.3) – (2.4), the Cauchy-Schwarz inequality (|u| , |v|) ≤ kuk kvk and
the non-negativeness of the following Gram determinant [3] or
kuk2
(|u| , |v|) y0 (2.5) 0 ≤ (|v| , |u|)
kvk2
x0 y0
x0
1 = kuk2 kvk2 − (|u| , |v|)2 − kuk2 x20 − 2(|u| , |v|)x0 y0 + kvk2 y02 ,
0 ≤ kuk2 kvk2 − (|u| , |v|)2 − A2
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John Michael Rassias
A = kuk x0 − kvk y0 ,
Z
x0 = |v(x)h(x)|dx,
ZI
y0 = |u(x)h(x)|dx,
ZI
kh(x)k2 = |h(x)|2 dx = 1,
I
(2.6) Mp∗ ≥ (|u| , |v|)2 =
I
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we find
Z
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2
2 Z
(p)
wp (x) f (x) f (x) dx ,
|u| |v| =
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I
where wp = (x − xm )p w. In general, if khk =
6 0, then one gets
(u, v)2 ≤ kuk2 kvk2 − R2 ,
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where
R = A/ khk = kuk x − kvk y,
such that x = x0 / khk , y = y0 / khk.
In this case, A has to be replaced by R in all the pertinent relations of this
paper.
From (2.6) and the complex inequality,
|ab| ≥
1
ab + ab
2
with a = wp (x) f (x), b = f (p) (x), we get
On the Refined
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John Michael Rassias
(2.7)
Z
2
1
∗
(p)
(p)
Mp =
wp (x)(f (x)f (x) + f (x)f (x))dx .
2 I
From (2.7) and the generalized differential identity (*), one finds
(2.8)


 2
Z
[p/2]
p−2q
X
2
1
d
Mp∗ ≥ 2  wp (x) 
Cq p−2q f (q) (x)  dx .
2
dx
I
q=0
From the generalized integral identity (1.2), the condition (2.1), 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)
wp(p−2q) (x) f (l) (x) dx = Iql .
dx
I
I
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Thus we find

[p/2]
Mp∗ ≥
1 
Cq
22 q=0
X
q
X
l=0
!2
1 2
Iql  = 2 Ep,I,f
,
2
P[p/2]
where Ep,I,f =
q=0 Cq Dq , if |Ep,I,f | < ∞ holds, or the refined moment
uncertainty formula
q
p
1 q
1
p
p 2p
∗
|Ep,I,f | ,
Mp ≥ √
Ep,I,f
≥ √
p
p
2
2
where Mp = Mp∗ + A2 .
We note that the corresponding Gram matrix to the above Gram determinant
is positive definite if and only if the above Gram determinant is positive if and
only if u, v, h are linearly independent. In addition, the equality in (2.5) holds if
and only if h is a linear combination of linearly independent u and v and u = 0
or v = 0, completing the proof of the above theorem.
On the Refined
Heisenberg-Weyl Type
Inequality
John Michael Rassias
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3.
Applied Refined Heisenberg-Weyl Type
Inequality
We apply the above Theorem 2.1 to the following simpler cases of the refined
Heisenberg-Weyl type inequality.
3.1.
Refined Second Moment Heisenberg-Weyl Type
Inequality
such that kf k22,I
2
R
2
For any f ∈ L (I), I = [0, ∞), f : I → C,
= I |f (x)| dx =
EI,|f |2 , any fixed but arbitrary constant xm ∈ R, and for the second order
moment
Z
2
(µ2 )I,|f |2 = σI,|f |2 = (x − xm )2 |f (x)|2 dx,
I
the second order moment Heisenberg-Weyl type inequality
Z
2
2 1
1
2
0 2
∗
2
∗
(R1 )
(µ2 )I,|f |2 · kf k2,I ≥
E
=
|f (x)| dx + 4A ,
4 1,I,f
4 I
holds, where E ∗1,I,f < ∞.
Equality holds in (R1∗ ) iff v(x) = −2c1 u(x) holds for constants c1 > 0,
and any fixed c1 = k12 /2 > 0, k1 ∈ R and k1 6= 0, and A = 0, or h(x) =
c11 u(x) + c21 v(x) and x0 = 0, or y0 = 0, where ci1 (i = 1, 2) are constants and
A2 > 0.
We note that this inequality (R1∗ ) still holds if we replace the interval of
integration I with R, without any other change except that one on the choice of
h, where (HI ) has to be replaced with (HR ).
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John Michael Rassias
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3.2.
Refined Fourth Moment Heisenberg-Weyl Type
Inequality
R
For any f ∈ L2 (I), I = [0, ∞), f : I → C, such that kf k22,I = I |f (x)|2 dx =
EI,|f |2 , any fixed but arbitrary constant xm ∈ R, and for the fourth order moment
Z
(µ4 )I,|f |2 = (x − xm )4 |f (x)|2 dx,
I
the fourth order moment Heisenberg-Weyl type inequality
1 ∗
2
(µ4 )I,|f |2 · kf 00 k2,I ≥ (E2,I,f
)2
4
Z h
2
i
1
2
2
2
0
2
=
|f (x)| dx − xδ |f (x)| dx + 4A
4 I
∗ < ∞.
holds, where xδ = x − xm , and E2,I,f
∗
Equality holds in (R2 ) iff v(x) = −2c2 u(x) holds for constants c2 > 0, and
any fixed but arbitrary c2 = 21 k22 > 0, k2 ∈ R and k2 6= 0, and A = 0, or
h(x) = c12 u(x) + c22 v(x) and x0 = 0, or y0 = 0, where ci2 (i = 1, 2) are
constants and A2 > 0.
We note that this inequality (R2∗ ) still holds if we replace the interval of
integration I with R, without any other change except that one on the above
condition (2.1), where x → ∞ has to be substituted with |x| → ∞, and the
choice of h, where (HI ) has to be replaced with (HR ).
(R2∗ )
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John Michael Rassias
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(p)
Remark 1. Take wp (x) = xp , and wp (x) = p! (p = 1, 2, 3, 4, . . .). Thus
Z
E1,I,f = − |f (x)|2 dx = −EI,|f |2 ,
Z Ih
i
2
E2,I,f = 2
|f (x)|2 − x2 |f 0 (x)| dx,
I
Z h
i
2
E3,I,f = −3
2 |f (x)|2 − 3x2 |f 0 (x)| dx,
Z hI
i
2
2
2
2
0
4
00
E4,I,f = 2
12 |f (x)| − 24x |f (x)| + x |f (x)| dx,
I
On the Refined
Heisenberg-Weyl Type
Inequality
John Michael Rassias
respectively, if |Ep,I,f | < ∞ holds for p = 1, 2, 3, 4. Therefore
Z
2
p−2q
wp(p−2q) (x) f (q) (x) dx,
Dq = Aqq Iqq = Iqq = (−1)
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if |Dq | < ∞, for 0 ≤ q ≤ p2 .
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J
Furthermore,
wp(p−2q) (x) = (xp )(p−2q) = p (p − 1) · · · (p − (p − 2q) + 1) xp−(p−2q) ,
II
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or
wp(p−2q) (x) =
p!
p! 2q
x2q =
x ,
(p − (p − 2q))!
(2q)!
0≤q≤
hpi
2
.
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In addition
Dq = (−1)p−2q
p!
(2q)!
Z
I
2
x2q f (q) (x) dx,
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if |Dq | < ∞ holds for 0 ≤ q ≤
Therefore
p
.
2
[p/2]
Ep,I,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)! I
or the formula
Z [p/2]
X
2
p!
p−q
p
x2q f (q) (x) dx,
Ep,I,f =
(−1)
·
p − q (2q)!
q
I q=0
if |Ep,I,f | < ∞ holds for 0 ≤ q ≤ p2 , when w = 1 and xm = 0.
Let
Z
(m2p )I,|f |2 = x2p |f (x)|2 dx
p−q
I
2
th
be the 2p moment of x for |f | about the origin xm = 0.
Denote
p
p!
p−q
p−q
εp,q = (−1)
·
,
p − q (2q)!
q
for p ∈ N and 0 ≤ q ≤ p2 .
Thus
Z [p/2]
X
2
Ep,I,f =
εp,q x2q f (q) (x) dx, if |Ep,I,f | < ∞
I q=0
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John Michael Rassias
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holds for 0 ≤ q ≤
p
.
2
Corollary 3.1. Assume that f : I → C is a complex valued function of a real
variable x, w = 1, xm = 0. If f ∈ L2 (I), then the following inequality
v
2
u
u[p/2]
q
q
X
1 u
p
p
2p
(p)
2
t
2 + 4A ,
ε
(m
)
(Sp )
(m2p )I,|f |2 kf k2,I ≥ √
p,q
2q
p
I,|f (q) | 2 q=0
holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 , where
Z
2
(m2q )I, f (q) 2 = x2q f (q) (x) dx
| |
I
and A is analogous to the one in the above theorem.
Similar conditions are assumed for the “equality” in (Sp ) with respect to
those in the above theorem. We note that this inequality (Sp ) still holds if we
replace the interval of integration I with R, without any other change except
that one on the above condition (2.1), where x → ∞ has to be substituted with
|x| → ∞, and the choice of h, where (HI ) has to be replaced with (HR ).
Problem 1. Concerning our inequality (H2 ) further investigation is needed for
the case of the “equality”. As a matter of fact, our function f is not in L2 (R),
leading the left-hand side to be infinite in that “equality”. A limiting argument
is required for this problem. On the other hand, why does not the corresponding
“inequality” (H2 ) attain an extremal in L2 (R)?
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John Michael Rassias
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Here are some of our old results [4] related to the above problem. In particular, if we take into account these results contained in Section 9 on pp. 46-70 [4],
where the Gaussian function and the Euler gamma function Γ are employed,
then via Corollary 9.1 on pp 50-51 of [4] we conclude that “equality” in (Hp ) of
[4, p. 22], p ∈ N = {1, 2, 3, . . .}, holds only for p = 1. Furthermore, employing
the above Gaussian function, we established the following extremum principle
(via (9.33) on p. 51 [4]):
(R)
R(p) ≥ 1/2π,
p∈N
for the corresponding “inequality” in (Hp ) of [4, p. 22], p ∈ N, where the constant 1/2π “on the right-hand side” is the best lower bound for p ∈ N. Therefore
“equality” in (Hp ) of [4, p. 22], p ∈ N and p 6= 1, in Section 8.1 on pp 19-46
[4] cannot occur under the afore-mentioned well-known functions. On the other
hand, there is a lower bound “on the right-hand side” of the corresponding “inequality” (H2 ) if we employ the above Gaussian function, which bound equals
4
1
1 |c0 |
2
to 64π
, with c0 , c constants and c0 ∈ C, c > 0, because
4 E2,R,f = 512π 3
c
2pπ
ER,|f |2 = |c0 |
and E2,R,f = 12 ER,|f |2 .
2c
Analogous pertinent results are investigated via our Corollaries 9.2-9.6 on
pp 53-68 [4].
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John Michael Rassias
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References
[1] B. BURKE HUBBARD, The World According to Wavelets, the Story
of a Mathematical Technique in the Making (A.K. Peters, Natick, Massachusetts, 1998).
[2] W. HEISENBERG, Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik, Zeit. Physik, 43, 172 (1927); The Physical Principles of the Quantum Theory (Dover, New York, 1949; The Univ.
Chicago Press, 1930).
[3] G. MINGZHE, On the Heisenberg’s inequality, J. Math. Anal. Appl., 234
(1999), 727–734.
[4] J.M. RASSIAS, On the Heisenberg-Pauli-Weyl inequality, J. Inequ. Pure
& Appl. Math., 5 (2004), Art. 4. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=356]
[5] J.M. RASSIAS, On the Heisenberg-Weyl inequality, J. Inequ. Pure &
Appl. Math., 6 (2005), Art. 11. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=480]
[6] H. WEYL, Gruppentheorie und Quantenmechanik, (S. Hirzel, Leipzig,
1928; and Dover edition, New York, 1950)
[7] N. WIENER, The Fourier Integral and Certain of its Applications, (Cambridge, 1933).
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[8] N. WIENER, I am a Mathematician, (MIT Press, Cambridge, 1956).
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