Journal of Inequalities in Pure and
Applied Mathematics
ON THE SHARPENED HEISENBERG-WEYL INEQUALITY
JOHN MICHAEL RASSIAS
Pedagogical Department, E. E.
National and Capodistrian University of Athens
Section of Mathematics and Informatics
4, Agamemnonos Str., Aghia Paraskevi
Athens 15342, Greece.
volume 7, issue 3, article 80,
2006.
Received 21 June, 2005;
accepted 21 July, 2006.
Communicated by: S. Saitoh
EMail: jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/∼jrassias/
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2000
Victoria University
ISSN (electronic): 1443-5756
188-05
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Abstract
The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that f : R → C is a complex
valued function of a random real variable x such that f ∈ L2 (R). Then the
product of the second moment of the random real x for |f|2 and the second
2
.
moment of the random real ξ for fˆ is at least E|f|2 4π, where fˆ is the Fourier
R
R
transform of f, such that fˆ (ξ) = R e−2iπξx f (x) dx, f (x) = R e2iπξx fˆ (ξ) dξ,
R
and E|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 higher order moments and in 2005, he investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized
Heisenberg-Weyl inequality is established in Fourier analysis. Afterwards, an
open problem is proposed on some pertinent extremum principle.These results
are useful in investigation of quantum mechanics.
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
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2000 Mathematics Subject Classification: 26, 33, 42, 60, 52.
Key words: Sharpened, Heisenberg-Weyl inequality, Gram determinant.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Second Order Moment Heisenberg-Weyl Inequality .
1.2
Fourth Order Moment Heisenberg-Weyl Inequality . .
2
Sharpened Heisenberg-Weyl Inequality . . . . . . . . . . . . . . . . . .
References
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1.
Introduction
The serious question of certainty in science was high-lighted by Heisenberg, in
1927, via his uncertainty principle [1]. He demonstrated 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], at 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 1928, according to
2 Pauli
[6] the less the uncertainty in |f |2 , the greater the uncertainty in fˆ , and
conversely. This result does not actually appear in Heisenberg’s seminal paper
[1] (in 1927). The following second order moment Heisenberg-Weyl inequality
provides a precise quantitative formulation of the above-mentioned uncertainty
principle according to W. Pauli.
1.1.
Second Order Moment Heisenberg-Weyl Inequality
([3, 4, 5]):
2
For any f ∈ L (R), f : R → C, such that
Z
2
kf k2 =
|f (x)|2 dx = E|f |2 ,
R
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John Michael Rassias
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any fixed but arbitrary constants xm , ξm ∈ R, and for the second order moments
Z
2
(µ2 )|f |2 = σ|f |2 =
(x − xm )2 |f (x)|2 dx,
ZR
2
2ˆ
2
(ξ − ξm ) f (ξ) dξ,
(µ2 ) fˆ 2 = σ ˆ 2 =
| |
f
| |
R
the second order moment Heisenberg-Weyl inequality
2
σ|f
|2
(H1 )
·
σ2ˆ 2
|f |
On The Sharpened
Heisenberg-Weyl Inequality
kf k42
≥
,
16π 2
John Michael Rassias
holds. Equality holds in (H1 ) if and only if the generalized Gaussians
f (x) = c0 exp (2πixξm ) exp −c (x − xm )2
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hold for some constants c0 ∈ C and c > 0.
1.2.
Fourth Order Moment Heisenberg-Weyl Inequality
([3, pp. 26-27]):
kf k22
2
R
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For any f ∈ L (R), f : R → C, such that
= R |f (x)| dx = E|f |2 , any
fixed but arbitrary constants xm , ξm ∈ R, and for the fourth order moments
Z
(µ4 )|f |2 =
(x − xm )4 |f (x)|2 dx
and
ZR
2
(µ4 ) fˆ 2 =
(ξ − ξm )4 fˆ(ξ) dξ,
| |
R
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the fourth order moment Heisenberg-Weyl inequality
1
2
E2,f
,
(µ4 )|f |2 · (µ4 ) fˆ 2 ≥
4
| |
64π
(H2 )
holds, where
Z h
i
2
2 2
(1−4π 2 ξm
xδ ) |f (x)|2 − x2δ |f 0 (x)| − 4πξm x2δ Im(f (x)f 0 (x)) dx,
E2,f = 2
R
with xδ = x − xm , ξδ = ξ − ξm , Im (·) is the imaginary part of (·), and
|E2,f | < ∞.
The “inequality” (H2 ) holds, unless f (x) = 0.
We note that if the ordinary differential equation of second order
(ODE)
John Michael Rassias
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fα00 (x) = −2c2 x2δ fα (x)
αx
On The Sharpened
Heisenberg-Weyl Inequality
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1 2
k
2 2
holds, with α = −2πξm i, fα (x) = e f (x), and a constant c2 =
> 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
2
2
2πixξm
|k2 | xδ + c21 J1/4
|k2 | xδ
,
f (x) = |xδ |e
c20 J−1/4
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 [3].
It is open to investigate cases, where the integrand on the right-hand side
of integral of E2,f will be nonnegative. For instance, for xm = ξm = 0, this
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integrand is: = |f (x)|2 − x2 |f 0 (x)|2 (≥ 0). In 2004, we ([3, 4]) generalized the
Heisenberg-Weyl inequality and in 2005 we [5] investigated a Heisenberg-Weyl
type inequality without Fourier transforms. In this paper, a sharpened form of
this generalized Heisenberg-Weyl inequality is established in Fourier analysis.
We state our following two pertinent propositions. For their proofs see [3].
Proposition 1.1 (Generalized differential identity, [3]).
If f : R → C is a
k
complex valued function of a real variable x, 0 ≤ 2 is the greatest integer
dj
≤ 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
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
k
(−1)
k−i
i
k−i
i
2
dk−2i (i)
,
f
(x)
dxk−2i
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤
i ∈ N0 = {0, 1, 2, . . .}.
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k
2
for
Proposition 1.2 (Lagrange type differential identity, [3]). If f : R → C is a
complex valued
function of a real variable x, and fa = eax f , where a = −βi,
√
with i = −1 and β = 2πξm for any fixed but arbitrary real constant ξm , as
well as if
p 2
Apk =
β 2(p−k) , 0 ≤ k ≤ p,
k
and
p p
Bpkj = spk
β 2p−j−k , 0 ≤ k < j ≤ p,
k
j
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where spk = (−1)p−k (0 ≤ k ≤ p), then
(LD)
p
X
2
(p) 2 X
fa =
Apk f (k) + 2
Bpkj Re rpkj f (k) f (j) ,
k=0
0≤khj≤p
holds for any fixed but arbitrary p ∈ N0 , where (·) is the conjugate of (·), and
k+j
rpkj = (−1)p− 2 (0 ≤ k < j ≤ p), and Re (·) is the real part of (·).
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Heisenberg-Weyl Inequality
John Michael Rassias
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2.
Sharpened Heisenberg-Weyl Inequality
We assume that f : R → C is a complex valued function of a real variable x (or
absolutely continuous in [−a, a], a > 0), and w : R → R a real valued weight
function of x, as well as xm , ξm any fixed√but arbitrary real constants. Denote
fa = eax f , where a = −2πξm i with i = −1, and fˆ the Fourier transform of
f , such that
Z
Z
−2iπξx
fˆ (ξ) =
e
f (x) dx and f (x) =
e2iπξx fˆ(ξ) dξ.
R
R
Also we denote
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
Z
w2 (x) (x − xm )2p |f (x)|2 dx,
ZR
2
(µ2p ) fˆ 2 =
(ξ − ξm )2p fˆ(ξ) dξ
| |
R
(µ2p )w,|f |2 =
2
the 2pth weighted moment
of
2 x for |f | with weight function w : R → R and
the 2pth moment of ξ for fˆ , respectively. In addition, we denote
hpi p−q
p
p
Cq = (−1)
, if 0 ≤ q ≤
= the greatest integer ≤
,
p−q
q
2
2
Z
hpi
2
Iql = (−1)p−2q wp(p−2q) (x) f (l) (x) dx, if 0 ≤ l ≤ q ≤
,
2
R
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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
,
k+j
where rqkj = (−1)q− 2 ∈ {±1, ±i} and wp = (x − xm )p w. We assume that
all these integrals exist. Finally we denote
Dq =
q
X
l=0
Aql =
β
l
2(q−l)
,
Bqkj = sqk
with β = 2πξm , and sqk = (−1) , and Ep,f
holds for p ∈ N.
In addition, we assume the two conditions:
q q β 2q−j−k ,
k
j
P
= [p/2]
q=0 Cq Dq , if |Ep,f | < ∞
r
(−1) lim
|x|→∞
r=0
wp(r)
2 (p−2q−r−1)
(l)
(x) f (x)
= 0,
r=0
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p−2q−1
(2.2)
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for 0 ≤ l ≤ q ≤ p2 , and
X
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p−2q−1
(2.1)
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p
, where
2
q−k
X
Bqkj Iqkj ,
0≤khj≤q
if |Dq | < ∞ holds for 0 ≤ q ≤
q 2
X
Aql Iql + 2
r
(−1) lim
|x|→∞
wp(r)
(x) Re rqkj f
(k)
(x) f
(j)
(p−2q−r−1)
(x)
= 0,
J. Ineq. Pure and Appl. Math. 7(3) Art. 80, 2006
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for 0 ≤ k < j ≤ q ≤
p
. Also,
2
q
∗
2
|Ep,f
| = Ep,f
+ 4A2 (≥ |Ep,f |),
where A =R kuk x0 − kvk y0 , with L2 −norm k·k2 =
(|u| , |v|) = R |u| |v|, and
R
R
|·|2 , inner product
u = w(x)xpδ fα (x),
v = fα(p) (x);
Z
Z
x0 =
|ν(x)| |h(x)| dx,
y0 =
|u(x)| |h(x)| dx,
R
R
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
as well as
2
1
− 1 x−µ
h(x) = √
√ e 4( σ ) ,
4
2π σ
where µ is the mean and σ the standard deviation, or
s
Γ n+1
1
1
2
h(x) = √
·
,
n+1
4
nπ
Γ n2
x2
4
1+ n
where n ∈ N, and
2
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kh(x)k =
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2
|h(x)| dx = 1.
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Theorem 2.1. If f ∈ L2 (R) (or absolutely continuous in [−a, a], a > 0), then
q
q
1 q
p ∗
∗ 2p
√
(Hp )
(µ2p )w,|f |2 2p (µ2p ) fˆ 2 ≥
E
p,f ,
| |
2π p 2
holds for any fixed but arbitrary p ∈ N.
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Equality holds in (Hp∗ ) 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.
Proof. In fact, from the generalized Plancherel-Parseval-Rayleigh identity [3,
(GPP)], and the fact that |eax | = 1 as a = −2πξm i, one gets
(2.3)
Mp∗
On The Sharpened
Heisenberg-Weyl Inequality
1
= Mp −
A2
2p
(2π)
John Michael Rassias
1
· (µ2p ) fˆ 2 −
A2
| |
(2π)2p
= (µ2p )w,|f |2
Z
2p
2
2
=
w (x) (x − xm ) |f (x)| dx
R
Z
2 1
2p ˆ
·
(ξ − ξm ) f (ξ) dξ −
A2
2p
(2π)
R
Z
1
=
w2 (x) (x − xm )2p |fa (x)|2 dx
2p
(2π)
R
Z
(p)
2
2
·
fa (x) dx − A
R
(2.4)
=
1 kuk2 kvk2 − A2
2p
(2π)
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(p)
with u = w(x)xpδ fα (x), v = fα (x).
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From (2.3) – (2.4), the Cauchy-Schwarz inequality (|u| , |v|) ≤ kuk kvk and
the non-negativeness of the following Gram determinant [2]
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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Heisenberg-Weyl Inequality
with
John Michael Rassias
Z
A = kuk x0 − kvk y0 , x0 =
Z
|ν(x)| |h(x)| dx, y0 =
R
and
2
|u(x)| |h(x)| dx,
R
Z
kh(x)k =
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2
|h(x)| dx = 1,
R
we find
(2.6)
1
(|u| , |v|)2
(2π)2p
Z
2
1
=
|u| |v|
(2π)2p
R
Z
2
1
(p)
wp (x) fa (x) fa (x) dx ,
=
(2π)2p
R
Mp∗ ≥
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where wp = (x − xm )p w, and fa = eax f . In general, if khk =
6 0, then one gets
(u, v)2 ≤ kuk2 kvk2 − R2 ,
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| ≥ 12 ab + ab with a =
(p)
wp (x) fa (x), b = fa (x), we get
Z
2
1
1
(p)
∗
(p)
(2.7)
Mp =
wp (x) fα (x)fα (x) + fα (x)fα (x) dx .
(2π)2p 2 R
From (2.7) and the generalized differential identity (*), one finds


 2
Z
[p/2]
p−2q X
2
1
d
(2.8)
Mp∗ ≥ 2(p+1) 2p  wp (x) 
Cq p−2q fa(q) (x)  dx .
2
π
dx
q=0
R
From (2.8) and the Lagrange type differential identity (LD), we find


Z
[p/2]
q
p−2q
X
X
2
1
d
∗
Mp ≥ 2(p+1) 2p  wp (x) 
Cq p−2q
Aql f (l) (x)
2
π
dx
R
q=0
l=0
 2
X
+2
Bqkj Re rqkj f (k) (x) f (j) (x)  dx .
0≤khj≤q
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From the generalized integral identity [3], the two conditions (2.1) – (2.2), 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
R
R
as well as
Z
dp−2q
wp (x) p−2q Re rqkj f (k) (x) f (j) (x)
dx
R
Z
p−2q
(p−2q)
(k)
(j)
= (−1)
wp
(x)Re rqkj f (x) f (x) = Iqkj .
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
R
Thus we find
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
Mp∗
≥
=
1
22(p+1) π 2p
1
22(p+1) π 2p
[p/2]

X
Cq 

q=0
q
X
l=0
2
Aql Iql + 2
X
Bqkj Iqkj 
0≤khj≤q
2
Ep,f
,
P[p/2]
where Ep,f =
q=0 Cq Dq , if |Ep,f | < ∞ holds, or the sharpened moment
uncertainty formula
q
p
1 q
1
p
p 2p
∗
√
√
Ep,f ≥
|Ep,f | ,
Mp ≥
2π p 2
2π p 2
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where Mp = Mp∗ +
1
A2 .
(2π)2p
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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. Besides, 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.
Let
Z
(m2p )|f |2 =
x2p |f (x)|2 dx
R
th
2
be the 2p moment of x for |f | about the origin xm = 0, and
Z
2
2p ˆ
(m2p ) fˆ 2 =
ξ f (ξ) dξ
| |
R
2
the 2pth moment of ξ for fˆ about the origin ξm = 0. Denote
p
p!
p−q
p−q
εp,q = (−1)
·
,
p − q (2q)!
q
if p ∈ N and 0 ≤ q ≤ p2 .
Corollary 2.2. 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 our theorem. If f : R → C (or absolutely continuous in [−a, a], a > 0), then
the following inequality
v
2
u
[p/2]
u
q
q
X
u
1
p
2p
t
√
(Sp )
(m2p )|f |2 2p (m2p ) fˆ 2 ≥
εp,q (m2q ) f (q) 2 + 4A2 ,
p
| |
| |
2π 2 q=0
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holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 , where
Z
2
(m2q ) f (q) 2 =
x2q f (q) (x) dx
| |
R
and A is analogous to the one in the above theorem.
We consider the extremum principle (via (9.33) on p. 51 of [3]):
1
,
p∈N
2π
for the corresponding “inequality” (Hp ) [3, p. 22], p ∈ N.
Problem 1. Employing our Theorem 8.1 on p. 20 of [3], the Gaussian function,
the Euler gamma function Γ, and other related special functions, we established
and explicitly proved the above extremum principle (R), where
Γ p + 12
,
R(p) = P
[p/2]
p
p!
p−q
Γq q=0 (−1) p−q · (2q)! p−q
q
R(p) ≥
(R)
with
[q/2]
X
q 2 1
1
2
2
Γ k+
Γq =
Γ 2q − 2k +
2k
2
2
k=0
q q X
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
On The Sharpened
Heisenberg-Weyl Inequality
John Michael Rassias
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p!
0 ≤ 2q is the greatest integer ≤ 2q for q ∈ N ∪ {0} = N0 , pq = q!(p−q)!
for
p ∈ N, q ∈ N0 and 0 ≤ q ≤ p, p! = 1 · 2 · 3 · · · · · (p − 1) · p and 0! = 1, as well
as
√
1
1
1 (2p)! √
Γ p+
= 2p ·
π, p ∈ N and Γ
= π.
2
2
p!
2
Furthermore, by employing computer techniques, this principle was verified
for p = 1, 2, 3, . . ., 32, 33, as well. It now remains open to give a second explicit
proof of verification for the extremum principle (R) using only special functions
techniques and without applying our Heisenberg-Pauli-Weyl inequality [3].
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References
[1] 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).
[2] G. MINGZHE, On the Heisenberg’s inequality, J. Math. Anal. Appl., 234
(1999), 727–734.
[3] 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].
[4] 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].
[5] J.M. RASSIAS, On the refined Heisenberg-Weyl type inequality, J. Inequ.
Pure & Appl. Math., 6 (2005), Art. 45. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=514].
[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).
J. Ineq. Pure and Appl. Math. 7(3) Art. 80, 2006
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