Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 2, Article 45, 2005
ON THE REFINED HEISENBERG-WEYL TYPE INEQUALITY
JOHN MICHAEL RASSIAS
NATIONAL AND C APODISTRIAN U NIVERSITY OF ATHENS
P EDAGOGICAL D EPARTMENT EE
S ECTION OF M ATHEMATICS ,4, AGAMEMNONOS S TR .
AGHIA PARASKEVI
ATHENS 15342, G REECE
jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/∼jrassias/
Received 11 January, 2005; accepted 17 March, 2005
Communicated by S. Saitoh
A BSTRACT. 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 mo 2
2
ment of the random real x for |f | and the second moment of the random real ξ for fb is
.
R
at least ER,|f |2 4π, where fb is the Fourier transform of f , fb(ξ) = R e−2iπξx f (x) dx and
R
R
2
f (x) = R e2iπξx fˆ (ξ) dξ, and ER,|f |2 = R |f (x)| dx. This uncertainty relation is wellknown 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.
Key words and phrases: Heisenberg-Weyl Type Inequality, Uncertainty Principle, Gram determinant.
2000 Mathematics Subject Classification. 26, 33, 42, 60, 62.
1. I NTRODUCTION
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]. In1928, according to Pauli [6] “ the less
2
the uncertainty in |f |2 , the greater the uncertainty in fb , and conversely.” This result does not
actually appear in Heisenberg’s seminal paper [2] (in 1927).
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
013-05
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J OHN M ICHAEL R ASSIAS
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
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 =
| |
Z
σ2 b 2
R,|f |
=
R
2
(ξ − ξm )2 fb(ξ) dξ,
the second order moment Heisenberg-Weyl inequality
(H1 )
2
σR,|f
|2
·
σ2 b 2
R,|f |
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 ofa signal cannot be less than the value 1/4π,
.
2
where ∆x2 = σR,|f
ER,|f |2 and ∆ξ 2 = σ 2 b 2 ER,|f |2 with f : R → C, fb : R → C defined
|2
R,|f |
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.
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
R
and
Z
(µ4 )R, fb 2 =
| |
R
b 2
(ξ − ξm ) f (ξ) dξ,
4
the fourth order moment Heisenberg-Weyl inequality
(H2 )
1
(µ4 )R,|f |2 · (µ4 )R, fb 2 ≥
E2 ,
| |
64π 4 2,R,f
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
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O N THE R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
3
holds, where
E2,R,f = 2
Z h
1−
2 2
xδ
4π 2 ξm
2
|f (x)| −
x2δ
0
2
|f (x)| −
4πξm x2δ
Im
i
f (x)f 0 (x)
dx,
R
with xδ = x − xm , ξδ = ξ − ξm , Im(·) is the imaginary part of (·), and |E2,R,f | < ∞.
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 = 21 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
2
2
2πixξm
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
f (x) = |xδ |e
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).
1.3. Second Moment Heisenberg-Weyl
Type Inequality ([4]). For any f ∈ L2 (I), I =
R
[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 2
1
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.
2
1.4. Fourth Moment Heisenberg-Weyl
R Type 2Inequality ([4]). For any f ∈ L (I), I =
2
[0, ∞), f : I → C, such that kf k2,I = I |f (x)| 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
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
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J OHN M ICHAEL R ASSIAS
the fourth order moment Heisenberg – Weyl type inequality
Z h
i 2
1 2
2
2
00 2
2
0
|f (x)| dx − xδ |f (x)| dx
(R2 )
(µ4 )I,|f |2 · kf k2,I ≥ E2,I,f =
4
I
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
k−i
k−1 k−i
k+1
k−i+1
(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 i ∈ N0 =
k
{0, 1, 2, . . .} such that −1
= 0.
Proposition 1.2 (Generalized differential identity,
valued
k [4]). If f : I → C isk a complex
dj
(j)
function of a real variable x, I = [0, ∞), 0 ≤ 2 is the greatest integer ≤ 2 , f = dxj f , and
(·) is the conjugate of (·), then
[ k2 ]
X
2
dk−2i (i)
f (x) ,
k−2i
dx
i=0
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤ k2 for i ∈ N0 =
{0, 1, 2, . . .}.
(*)
f (x) f
(k)
(x) + f
(k)
(x) f¯ (x) =
k
(−1)
k−i
i
k−i
i
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 product f1 f2 is given, in
(m)
(p−m)
terms of the lower derivatives f1 , f2
by
p X
p (m) (p−m)
(p)
f f
(1.1)
(f1 f2 ) =
m 1 2
m=0
for any fixed but arbitrary p ∈ N0 .
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,
p
wp : I → R are two real valued functions of x, such that w
pp(x) = (x − xm ) w(x) for any fixed
but arbitrary constant xm ∈ R and v = p − 2q, 0 ≤ q ≤ 2 , then
i)
Z
Z
v−1
X
r (r)
v
(v)
(v−r−1)
(1.2)
wp (x) h (x) dx =
(−1) wp (x) h
(x) + (−1)
wp(v) (x) h (x) dx
r=0
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O N THE R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
5
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
(−1)r lim wp(r) (x) h(v−r−1) (x) = 0,
x→∞
r=0
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| → ∞.
2. R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
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
2
th
the 2p weighted moment of x for |f | with weight function w : I → R. Besides we denote
p−q
p
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 exist. Finally
Pq
p
we denote Dq = l=0 Iql , if |Dq | < ∞ holds for 0 ≤ q ≤ 2 , and
[p/2]
Ep,I,f =
X
Cq Dq ,
q=0
if |Ep,I,f | < ∞ holds for p ∈ N. In addition, we assume the condition:
p−2q−1
(2.1)
X
r=0
for 0 ≤ l ≤ q ≤
2 (p−2q−r−1)
(−1)r lim wp(r) (x) f (l) (x)
= 0,
x→∞
p
. Furthermore,
2
∗
|Ep,I,f
|=
q
2
Ep,I,f
+ 4A2 ,
R
R
where A = kuk x0 − kvk y0 , with L2 −norm k·k2 = I |·|2 , inner product (|u| , |v|) = I |u| |v|,
and
Z
Z
p
(p)
u = w(x)xδ f (x), v = f (x); x0 = |v(x)h(x)|dx, y0 = |u(x)h(x)|dx,
(2.2)
I
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
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J OHN M ICHAEL R ASSIAS
as well as
1
h(x) = √
σ
r
4
2 − 1 ( x−µ )2
e 4 σ ,
π
or
h(x) =
(HI )
√
1
2√
4
nπ
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, and
Z
2
kh(x)k = |h(x)|2 dx = 1.
I
2
Theorem 2.1. If (2.1) holds and f ∈ L (R), then
q
q
1 q
∗
p
∗
2p
(p)
,
√
(µ2p )w,I,|f |2 kf k2,I ≥ p p Ep,I,f
(Rp )
2
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
1
− 14 ( x−µ
)2
σ
h(x) = √
e
,
√
4
2π σ
or
s
Γ n+1
1
1
2
(HR )
h(x) = √
·
n+1 ,
n
4
2
nπ
Γ 2
1 + xn 4
where µ is the mean, σ the standard deviation, and n ∈ N.
Proof. In fact, one gets
(2.3)
Mp∗ = Mp − A2
2
= (µ2p )w,I,|f |2 · f (p) 2,I − A2
Z
Z
(p) 2
2p
2
2
f (x) dx − A2
=
w (x) (x − xm ) |f (x)| dx ·
(2.4)
I
2
I
2
= kuk kvk − A
2
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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O N THE R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
7
with
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
we find
(2.6)
Mp∗
2
Z
≥ (|u| , |v|) =
I
2 Z
2
(p)
|u| |v| =
wp (x) f (x) f (x) dx ,
I
p
where wp = (x − xm ) w. 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,
1
|ab| ≥
ab + ab
2
(p)
with a = wp (x) f (x), b = f (x), we get
2
Z
1
∗
(p)
(p)
(2.7)
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
Z
[p/2]
p−2q
X
2
1
d
(2.8)
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
Thus we find

!2
[p/2]
q
X
X
1
1 2
Mp∗ ≥ 2 
Cq
Iql  = 2 Ep,I,f
,
2 q=0
2
l=0
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
∗
Mp ≥ √
Ep,I,f
≥ √
|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
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8
J OHN M ICHAEL R ASSIAS
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.
3. A PPLIED R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
We apply the above Theorem 2.1 to the following simpler cases of the refined HeisenbergWeyl type inequality.
3.1. Refined Second Moment Heisenberg-Weyl
Type Inequality. For any f ∈ L2 (I), I =
R
[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
2
0
∗
2
(µ2 )I,|f |2 · kf k2,I ≥
E
=
|f (x)| dx + 4A ,
(R1∗ )
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 ).
3.2. Refined Fourth Moment Heisenberg-Weyl
Type Inequality. For any f ∈ L2 (I), I =
R
[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
2
Z h
i
1
1
2
2
2
00
2
0
2
∗
2
(µ4 )I,|f |2 · kf k2,I ≥ (E2,I,f ) =
(R2∗ )
|f (x)| dx − xδ |f (x)| dx + 4A
4
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 = 12 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 ).
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
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O N THE R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
9
(p)
Remark 3.1. 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
E4,I,f = 2
12 |f (x)|2 − 24x2 |f 0 (x)| + x4 |f 00 (x)| dx,
I
respectively, if |Ep,I,f | < ∞ holds for p = 1, 2, 3, 4. Therefore
Z
2
p−2q
Dq = Aqq Iqq = Iqq = (−1)
wp(p−2q) (x) f (q) (x) dx,
I
p
if |Dq | < ∞, for 0 ≤ q ≤ 2 .
Furthermore,
wp(p−2q) (x) = (xp )(p−2q) = p (p − 1) · · · (p − (p − 2q) + 1) xp−(p−2q) ,
or
hpi
p!
p! 2q
x2q =
x , 0≤q≤
.
(p − (p − 2q))!
(2q)!
2
In addition
Z
2
p−2q p!
x2q f (q) (x) dx,
Dq = (−1)
(2q)! I
p
if |Dq | < ∞ holds for 0 ≤ q ≤ 2 .
Therefore
Z
[p/2]
[p/2] X
X
2
p
p−q
q
p−2q p!
2q (q)
Ep,I,f =
Cq Dq =
(−1)
(−1)
x f (x) dx ,
p−q
q
(2q)! I
q=0
q=0
wp(p−2q) (x) =
or the formula
Ep,I,f =
Z [p/2]
X
p−q
(−1)
I q=0
p
p!
·
p − q (2q)!
p−q
q
2
x2q f (q) (x) dx,
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
I
th
2
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
holds for 0 ≤ q ≤ p2 .
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
http://jipam.vu.edu.au/
10
J OHN M ICHAEL R ASSIAS
Corollary 3.2. 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)
t
(Sp )
(m2p )I,|f |2 kf k2,I ≥ √
εp,q (m2q )I, f (q) 2 + 4A2 ,
p
| |
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)?
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
pπ
4
1
1 |c0 |
2
to 64π
, with c0 , c constants and c0 ∈ C, c > 0, because ER,|f |2 = |c0 |2 2c
4 E2,R,f = 512π 3
c
and E2,R,f = 12 ER,|f |2 .
Analogous pertinent results are investigated via our Corollaries 9.2-9.6 on pp 53-68 [4].
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
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O N THE R EFINED H EISENBERG -W EYL T YPE I NEQUALITY
11
[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).
[8] N. WIENER, I am a Mathematician, (MIT Press, Cambridge, 1956).
J. Inequal. Pure and Appl. Math., 6(2) Art. 45, 2005
http://jipam.vu.edu.au/