HEISENBERG-PAULI-WEYL UNCERTAINTY
PRINCIPLE FOR THE RIEMANN-LIOUVILLE
OPERATOR
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
S. OMRI AND L.T. RACHDI
vol. 9, iss. 3, art. 88, 2008
Department of Mathematics
Faculty of Sciences of Tunis
2092 Manar 2 Tunis, Tunisia.
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EMail: lakhdartannech.rachdi@fst.rnu.tn
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Received:
29 March, 2008
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Accepted:
08 August, 2008
Page 1 of 45
Communicated by:
J.M. Rassias
2000 AMS Sub. Class.:
42B10, 33C45.
Key words:
Heisenberg-Pauli-Weyl Inequality, Riemann-Liouville operator, Fourier transform, local uncertainty principle.
Abstract:
The Heisenberg-Pauli-Weyl inequality is established for the Fourier transform
connected with the Riemann-Liouville operator.Also, a generalization of this inequality is proved. Lastly, a local uncertainty principle is studied.
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Contents
1
Introduction
3
2
The Fourier Transform Associated with the Riemann-Liouville Operator
8
3
Hilbert Basis of the Spaces L2 (dνα ), and L2 (dγα )
14
4
Heisenberg-Pauli-Weyl Inequality for the Fourier Transform Fα
21
5
The Local Uncertainty Principle
30
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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1.
Introduction
Uncertainty principles play an important role in harmonic analysis and have been
studied by many authors and from many points of view [8].These principles state
that a function f and its Fourier transform fb cannot be simultaneously sharply localized. The theorems of Hardy, Morgan, Beurling, ... are established for several
Fourier transforms in [4], [9], [13] and [14]. In this context, a remarkable Heisenberg uncertainty principle [10] states, according to Weyl [25] who assigned the result
to Pauli, that for all square integrable functions f on Rn with respect to the Lebesgue
measure, we have
Z
Z
Z
2
1
2
2
2 b
2
2
xj |f (x)| dx
ξj |f (ξ)| dξ ≥
|f (x)| dx , j ∈ {1, . . . , n}.
4
Rn
Rn
Rn
This inequality is called the Heisenberg-Pauli-Weyl inequality for the classical Fourier
transform.
Recently, many works have been devoted to establishing the Heisenberg-PauliWeyl inequality for various Fourier transforms, Rösler [21] and Shimeno [22] have
proved this inequality for the Dunkl transform, in [20] Rösler and Voit have established an analogue of the Heisenberg-Pauli-Weyl inequality for the generalized Hankel transform. In the same context, Battle [3] has proved this inequality for wavelet
states, and Wolf [26], has studied this uncertainty principle for Gelfand pairs. We
cite also De Bruijn [5] who has established the same result for the classical Fourier
transform by using Hermite Polynomials, and Rassias [17, 18, 19] who gave several
generalized forms for the Heisenberg-Pauli-Weyl inequality.
In [2], the second author with others considered the singular partial differential
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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operators defined by

∂
∆1 = ∂x
,
∆ =
2
∂2
∂r2
+
2α+1 ∂
r ∂r
−
∂2
;
∂x2
(r, x) ∈]0, +∞[×R; α ≥ 0.
and they associated to ∆1 and ∆2 the following integral transform, called the RiemannLiouville operator, defined on C∗ (R2 ) (the space of continuous functions on R2 , even
with respect to the first variable) by
Rα (f )(r, x)
√
 α R1 R1
1
 π −1 −1 f rs 1 − t2 , x + rt (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
=
√
R
 1 1 f r 1 − t2 , x + rt √ dt ;
if α = 0.
π −1
2
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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Contents
(1−t )
In addition, a convolution product and a Fourier transform Fα connected with the
mapping Rα have been studied and many harmonic analysis results have been established for the Fourier transform Fα (Inversion formula, Plancherel formula, PaleyWiner and Plancherel theorems, ...).
Our purpose in this work is to study the Heisenberg-Pauli-Weyl uncertainty principle for the Fourier transform Fα connected with Rα .More precisely, using Laguerre and Hermite polynomials we establish firstly the Heisenberg-Pauli-Weyl inequality for the Fourier transform Fα , that is
• For all f ∈ L2 (dνα ), we have
Z
0
+∞
Z
R
12
(r + x )|f (r, x)| dνα (r, x)
2
2
2
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12
(µ + 2λ )|Fα (f )(µ, λ)| dγα (µ, λ)
Γ+
Z +∞ Z
2α + 3
2
≥
|f (r, x)| dνα (r, x) ,
2
0
R
Z Z
×
2
2
2
with equality if and only if
f (r, x) = Ce
2
2
− r +x
2t2
0
Heisenberg-Pauli-Weyl
Uncertainty Principle
; C ∈ C, t0 > 0,
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
where
• dνα (r, x) is the measure defined on R+ × R by
dνα (r, x) =
r
2α Γ(α
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2α+1
√ dr ⊗ dx.
+ 1) 2π
Contents
• dγα (µ, λ) is the measure defined on the set
Γ+ = R+ × R ∪ (it, x); (t, x) ∈ R+ × R; t ≤ |x| ,
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by
Z Z
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g(µ, λ)dγα (µ, λ)
Γ+
1
√
=
2α Γ(α + 1) 2π
Z
+∞
0
Close
Z
g(µ, λ)(µ2 + λ2 )α µdµdλ
R
Z Z
!
|λ|
2
2 α
g(iµ, λ)(λ − µ ) µdµdλ .
+
R
0
Next, we give a generalization of the Heisenberg-Pauli-Weyl inequality, that is
• For all f ∈ L2 (dνα ), a, b ∈ R; a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, we
have
Z +∞ Z
η2
(r2 + x2 )a |f (r, x)|2 dνα (r, x)
0
R
Z Z
×
1−η
2
2
2 b
2
(µ + 2λ ) |Fα (f )(µ, λ)| dγα (µ, λ)
Γ+
≥
2α + 3
2
aη Z
0
+∞
Z
12
|f (r, x)| dνα (r, x) ,
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
2
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with equality if and only if
a = b = 1 and f (r, x) = Ce
−r
2 +x2
2t2
0
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;
C ∈ C;
t0 > 0.
In the last section of this paper, building on the ideas of Faris [7], and Price [15, 16],
we develop a family of inequalities in their sharpest forms, which constitute the
principle of local uncertainty.
Namely, we have established the following main results
• For all real numbers ξ; 0 < ξ < 2α+3
, there exists a positive constant Kα,ξ
2
2
such that for all f ∈ L (dνα ), and for all measurable subsets E ⊂ Γ+ ; 0 <
γα (E) < +∞, we have
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ)
E
Z +∞ Z
2ξ
2α+3
< Kα,ξ γα (E)
(r2 + x2 )ξ |f (r, x)|2 dνα (r, x).
0
R
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• For all real number ξ; ξ > 2α+3
, there exists a positive constant Mα,ξ such that
2
for all f ∈ L2 (dνα ), and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) <
+∞, we have
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ)
E
Z
+∞
Z
< Mα,ξ γα (E)
0
2ξ−2α−3
2ξ
|f (r, x)|2 dνα (r, x)
S. Omri and L.T. Rachdi
R
Z
+∞ Z
×
0
Heisenberg-Pauli-Weyl
Uncertainty Principle
2α+3
2ξ
2
2 ξ
2
(r + x ) |f (r, x)| dνα (r, x)
,
vol. 9, iss. 3, art. 88, 2008
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where Mα,ξ is the best (the smallest) constant satisfying this inequality.
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2.
The Fourier Transform Associated with the Riemann-Liouville
Operator
It is well known [2] that for all (µ, λ) ∈ C2 , the system


∆1 u(r, x) = −iλu(r, x),




∆2 u(r, x) = −µ2 u(r, x),




u(0, 0) = 1, ∂u (0, x) = 0, ∀x ∈ R,
∂r
admits a unique solution ϕµ,λ , given by
(2.1)
∀(r, x) ∈ R2 ;
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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p
ϕµ,λ (r, x) = jα r µ2 + λ2 e−iλx ,
where
+∞
x 2n
X
Jα (x)
(−1)n
jα (x) = 2 Γ(α + 1) α = Γ(α + 1)
,
x
n!Γ(α + n + 1) 2
n=0
α
and Jα is the Bessel function of the first kind and index α [6, 11, 12, 24].
The modified Bessel function jα has the following integral representation [1, 11],
for all µ ∈ C, and r ∈ R we have

R
 √2Γ(α+1)1 1 (1 − t2 )α− 12 cos(rµt)dt, if α > − 1 ;
2
πΓ(α+ 2 ) 0
jα (rµ) =
 cos(rµ),
if α = − 12 .
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In particular, for all r, s ∈ R, we have
Z 1
1
2Γ(α + 1)
|jα (rs)| ≤ √
(1 − t2 )α− 2 | cos(rst)|dt
1
πΓ α + 2 0
Z 1
1
2Γ(α + 1)
≤√
(1 − t2 )α− 2 dt = 1.
1
πΓ(α + 2 ) 0
(2.2)
From the properties of the Bessel function, we deduce that the eigenfunction ϕµ,λ
satisfies the following properties
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
•
(2.3)
sup |ϕµ,λ (r, x)| = 1,
(r,x)∈R2
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if and only if (µ, λ) belongs to the set
Γ = R2 ∪ {(it, x); (t, x) ∈ R2 ; |t| ≤ |x|}.
• The eigenfunction ϕµ,λ has the following Mehler integral representation
 R1 R1
√
1
 απ −1 −1 cos(µrs 1 − t2 )e−iλ(x+rt) (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
ϕµ,λ (r, x) =
 1 R 1 cos(rµ√1 − t2 )e−iλ(x+rt) √ dt ;
if α = 0.
π −1
1−t2
In [2], using this integral representation, the authors have defined the RiemannLiouville integral transform associated with ∆1 , ∆2 by
√
 α R1 R1
1
 π −1 −1 f rs 1 − t2 , x + rt (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
Rα (f )(r, x) =
√
R
 1 1 f r 1 − t2 , x + rt √ dt ;
if α = 0.
π −1
2
(1−t )
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where f is a continuous function on R2 , even with respect to the first variable.
The transform Rα generalizes the "mean operator" defined by
Z 2π
1
R0 (f )(r, x) =
f (r sin θ, x + r cos θ)dθ.
2π 0
In the following we denote by
Heisenberg-Pauli-Weyl
Uncertainty Principle
• dνα the measure defined on R+ × R, by
dνα (r, x) =
r2α+1
√ dr ⊗ dx.
2α Γ(α + 1) 2π
• Lp (dνα ) the space of measurable functions f on R+ × R such that
Z +∞ Z
p1
p
kf kp,να =
|f (r, x)| dνα (r, x)
< ∞, if p ∈ [1, +∞[,
0
R
kf k∞,να = ess sup(r,x)∈R+ ×R |f (r, x)| < ∞,
if p = +∞.
• h / iνα the inner product defined on L2 (dνα ) by
Z +∞ Z
hf /giνα =
f (r, x)g(r, x)dνα (r, x).
0
R
• Γ+ = R+ × R ∪ (it, x); (t, x) ∈ R+ × R; t ≤ |x| .
• BΓ+ the σ-algebra defined on Γ+ by
BΓ+ = {θ−1 (B), B ∈ B(R+ × R)},
where θ is the bijective function defined on the set Γ+ by
p
(2.4)
θ(µ, λ) =
µ2 + λ 2 , λ .
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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• dγα the measure defined on BΓ+ by
∀ A ∈ BΓ+ ; γα (A) = να (θ(A))
(2.5)
• Lp (dγα ) the space of measurable functions f on Γ+ , such that
Z Z
p1
p
kf kp,γα =
|f (µ, λ)| dγα (µ, λ) < ∞, if p ∈ [1, +∞[,
Γ+
kf k∞,γα = ess sup(µ,λ)∈Γ+ |f (µ, λ)| < ∞,
if p = +∞.
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
2
• h / iγα the inner product defined on L (dγα ) by
Z Z
hf /giγα =
f (µ, λ)g(µ, λ)dγα (µ, λ).
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Γ+
Then, we have the following properties.
Proposition 2.1.
i) For all non negative measurable functions g on Γ+ , we have
Z Z
(2.6)
g(µ, λ)dγα (µ, λ)
Γ+
Z +∞ Z
1
√
=
g(µ, λ)(µ2 + λ2 )α µdµdλ
α
2 Γ(α + 1) 2π
0
R
!
Z Z |λ|
2
2 α
+
g(iµ, λ)(λ − µ ) µdµdλ .
R
0
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In particular
dγα+1 (µ, λ) =
(2.7)
µ2 + λ 2
dγα (µ, λ).
2(α + 1)
ii) For all measurable functions f on R+ × R, the function f ◦ θ is measurable
on Γ+ . Furthermore, if f is non negative or an integrable function on R+ × R
with respect to the measure dνα , then we have
Z Z
Z +∞ Z
(2.8)
(f ◦ θ)(µ, λ)dγα (µ, λ) =
f (r, x)dνα (r, x).
Γ+
0
Definition 2.2. The Fourier transform Fα associated with the Riemann-liouville
operator Rα is defined on L1 (dνα ) by
Z +∞ Z
(2.9)
∀(µ, λ) ∈ Γ; Fα (f )(µ, λ) =
f (r, x)ϕµ,λ (r, x)dνα (r, x).
R
By the relation (2.3), we deduce that the Fourier transform Fα is a bounded linear
operator from L1 (dνα ) into L∞ (dγα ), and that for all f ∈ L1 (dνα ), we have
(2.10)
kFα (f )k∞,γα ≤ kf k1,να .
Γ+
vol. 9, iss. 3, art. 88, 2008
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Theorem 2.3 (Inversion formula). Let f ∈ L (dνα ) such that Fα (f ) ∈ L (dγα ),
then for almost every (r, x) ∈ R+ × R, we have
Z Z
f (r, x) =
Fα (f )(µ, λ)ϕµ,λ (r, x)dγα (µ, λ).
1
S. Omri and L.T. Rachdi
R
In the following, we shall define the Fourier transform Fα associated with the
operator Rα and we give some properties that we use in the sequel.
0
Heisenberg-Pauli-Weyl
Uncertainty Principle
1
Theorem 2.4 (Plancherel). The Fourier transform Fα can be extended to an isometric isomorphism from L2 (dνα ) onto L2 (dγα ).
In particular, for all f, g ∈ L2 (dνα ), we have the following Parseval’s equality
Z Z
(2.11)
Fα (f )(µ, λ)Fα (g)(µ, λ)dγα (µ, λ)
Γ+
Z
+∞
Heisenberg-Pauli-Weyl
Uncertainty Principle
Z
=
f (r, x)g(r, x)dνα (r, x).
0
R
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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3.
Hilbert Basis of the Spaces L2 (dνα ), and L2 (dγα )
In this section, using Laguerre and Hermite polynomials, we construct a Hilbert basis
of the spaces L2 (dνα ) and L2 (dγα ), and establish some intermediate results that we
need in the next section.
It is well known [11, 23] that for every α ≥ 0, the Laguerre polynomials Lαm are
defined by the following Rodriguez formula
Lαm (r) =
1 r −α dm m+α −r er
r
e ;
m!
drm
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
m ∈ N.
vol. 9, iss. 3, art. 88, 2008
Also, the Hermite polynomials are defined by the Rodriguez formula
Hn (x) = (−1)n ex
Moreover, the families
(s
)
m!
Lα
Γ(α + m + 1) m
2
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dn −x2 e
;
dxn
n ∈ N.
(s
and
m∈N
1
√ Hn
n
2 n! π
Contents
)
2
are respectively a Hilbert basis of the Hilbert spaces L2 (R+ , e−r rα dr) and L2 (R, e−x dx).
Therefore the families
(s
)
(s
)
x2
2α+1 Γ(α + 1)m! − r2 α 2
1
√ e− 2 Hn
e 2 Lm (r )
and
Γ(α + m + 1)
2n n! π
n∈N
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n∈N
m∈N
JJ
2α+1
are respectively a Hilbert basis of the Hilbert spaces L2 R+ , 2αrΓ(α+1) dr and L2 R, √dx2π ,
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n
o
α
hence the family em,n
(m,n)∈N2
defined by
2α+1 Γ(α + 1)m!
eαm,n (r, x) =
! 12
e−
1
2n− 2 n!Γ(m + α + 1)
r 2 +x2
2
Lαm (r2 )Hn (x),
is a Hilbert basis of the space L2 (dνα ).
n
o
α
Using the relation (2.8), we deduce that the family ξm,n
(m,n)∈N2
α
ξm,n
(µ, λ)
=
(eαm,n
=
, defined by
◦ θ)(µ, λ)
2α+1 Γ(α + 1)m!
1
2n− 2 n!Γ(m + α + 1)
e−
µ2 +2λ2
2
Lαm (µ2 + λ2 )Hn (λ),
Proposition 3.1. For all (m, n) ∈ N2 , (r, x) ∈ R+ × R and (µ, λ) ∈ Γ+ , we have
r
r
n+1 α
n α
α
(3.1)
xem,n (r, x) =
em,n+1 (r, x) +
e
(r, x).
2
2 m,n−1
(3.2)
α
λξm,n
(µ, λ)
(3.3) r2 eα+1
m,n (r, x) =
p
=
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
! 12
is a Hilbert basis of the space L2 (dγα ), where θ is the function defined by the relation
(2.4).
In the following, we agree that the Laguerre and Hermite polynomials with negative index are zero.
r
Heisenberg-Pauli-Weyl
Uncertainty Principle
n+1 α
ξ
(µ, λ) +
2 m,n+1
r
n α
ξ
(µ, λ).
2 m,n−1
2(α + 1)(α + m + 1)eαm,n (r, x)
p
− 2(α + 1)(m + 1)eαm+1,n (r, x).
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α+1
(3.4) (µ2 + λ2 )ξm,n
(µ, λ) =
p
α
2(α + 1)(α + m + 1)ξm,n
(µ, λ)
p
α
− 2(α + 1)(m + 1)ξm+1,n
(µ, λ).
Proof. We know [11] that the Hermite polynomials satisfy the following recurrence
formula
Hn+1 (x) − 2xHn (x) + 2nHn−1 (x) = 0; n ∈ N,
Therefore, for all (r, x) ∈ R+ × R, we have
! 12
α+1
2
2
2
Γ(α
+
1)m!
− r +x
2
e
Lαm (r2 )xHn (x)
xeαm,n (r, x) =
n− 12
2
n!Γ(m + α + 1)
! 12
r
2
2
n+1
2α+1 Γ(α + 1)m!
− r +x
2
e
Lαm (r2 )Hn+1 (x)
=
n+ 12
2
2
(n + 1)!Γ(m + α + 1)
! 12
r
|(r,x)|2
2α+1 Γ(α + 1)m!
n
− 2
+
e
Lαm (r2 )Hn−1 (x)
2 2n− 32 (n − 1)!Γ(m + α + 1)
r
r
n+1 α
n α
=
em,n+1 (r, x) +
e
(r, x)
2
2 m,n−1
α
and it is obvious that the same relation holds for the elements ξm,n
.
On the other hand
! 12
α+2
2
2
2
Γ(α
+
2)m!
− r +x
2
2
r2 eα+1
r2 Lα+1
e
1
m,n (r, x) =
m (r )Hn (x).
n− 2
2
n!Γ(m + α + 2)
However, the Laguerre polynomials satisfy the following recurrence formulas
(m + 1)Lαm+1 (r) + (r − α − 2m − 1)Lαm (r) + (m + α)Lαm−1 (r) = 0;
m ∈ N,
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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and
α+1
α
Lα+1
m (r) − Lm−1 (r) = Lm (r);
m ∈ N.
Hence, we deduce that
r2 eα+1
m,n (r, x)
=
2α+2 Γ(α + 2)m!
n− 12
2
2
−
=
p
Heisenberg-Pauli-Weyl
Uncertainty Principle
Hn (x)
− (m +
2
1)Lα+1
m+1 (r )
! 12
n!Γ(m + α + 2)
(α + m + 1)e−
2α+2 Γ(α + 2)m!
1
r 2 +x2
2
S. Omri and L.T. Rachdi
2
2)Lα+1
m (r )
2α+2 Γ(α + 2)m!
n− 12
e−
n!Γ(m + α + 2)
× (α + 2m +
=
! 12
2n− 2 n!Γ(m + α + 2)
! 12
r 2 +x2
2
− (α + m +
2
1)Lα+1
m−1 (r )
Lαm (r2 )Hn (x)
vol. 9, iss. 3, art. 88, 2008
Title Page
Contents
2
2
− r +x
2
(m + 1)e
2(α + 1)(α + m + 1)eαm,n (r, x) −
Lαm+1 (r2 )Hn (x)
p
2(α + 1)(m + 1)eαm+1,n (r, x).
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Proposition 3.2. For all (m, n) ∈ N2 , and (µ, λ) ∈ Γ+ , we have
(3.5)
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α
Fα (eαm,n )(µ, λ) = (−i)2m+n ξm,n
(µ, λ).
2
Proof. It is clear that for all (m, n) ∈ N , the function
eαm,n
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belongs to the space
L1 (dνα ) ∩ L2 (dνα ), hence by using Fubini’s theorem, we get
2α+1 Γ(α + 1)m!
Fα (eαm,n )(µ, λ) =
! 12
1
2n− 2 n!Γ(m + α + 1)
Z +∞
p
r2α+1
2
− r2 α
2
×
e Lm (r )jα r µ2 + λ2 α
dr
2 Γ(α + 1)
0
Z
2
dx
− x2 −iλx
Hn (x) √
×
e
,
2π
R
and then the required result follows from the following equalities [11]:
Z +∞
y
r
α
α
√
∀m ∈ N;
e− 2 Lαm (r)Jα ( ry)r 2 dr = (−1)m 2e− 2 y 2 Lαm (y),
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
0
and
Contents
√
y2
x2
eixy e− 2 Hn (x)dx = in 2πe− 2 Hn (y),
Z
∀n ∈ N;
R
where Jα denotes the Bessel function of the first kind and index α defined for all
x > 0 by
+∞
x 2n+α
X
(−1)n
.
Jα (x) =
n!Γ(α + n + 1) 2
n=0
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Proposition 3.3. Let f ∈ L (dνα ) ∩ L (dνα+1 ) such that Fα (f ) ∈ L (dγα+1 ), then
for all (m, n) ∈ N2 , we have
s
s
α
+
m
+
1
m+1
(3.6)
hf /eα+1
hf /eαm,n iνα −
hf /eαm+1,n iνα ,
m,n iνα+1 =
2(α + 1)
2(α + 1)
2
2
2
and
s
α+1
(3.7) hFα (f )/ξm,n
iγα+1 =
α+m+1
(−i)2m+n hf /eαm,n iνα
2(α + 1)
s
m+1
+
(−i)2m+n hf /eαm+1,n iνα .
2(α + 1)
Proof. We have
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
hf /eα+1
m,n iνα+1 =
Z
+∞ Z
f (r, x)eα+1
m,n (r, x)dνα+1 (r, x)
0
R
Z +∞ Z
1
f (r, x)r2 eα+1
=
m,n (r, x)dνα (r, x)
2(α + 1) 0
R
1
=
hf /r2 eα+1
m,n iνα ,
2(α + 1)
hence by using the relation (3.3), we deduce that
s
s
α
+
m
+
1
m+1
hf /eα+1
hf /eαm,n iνα −
hf /eαm+1,n iνα .
m,n iνα+1 =
2(α + 1)
2(α + 1)
In the same manner, and by virtue of the relation (2.7), we have
Z Z
α+1
α+1
hFα (f )/ξm,n iγα+1 =
Fα (f )(µ, λ)ξm,n
(µ, λ)dγα+1 (µ, λ)
Γ+
Z Z
1
α+1
=
Fα (f )(µ, λ)(µ2 + λ2 )ξm,n
(µ, λ)dγα (µ, λ),
2(α + 1)
Γ+
vol. 9, iss. 3, art. 88, 2008
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Contents
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Page 19 of 45
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using the relations (3.4) and (3.5), we deduce that
Z Z
√
1
α+1
α
Fα (f )(µ, λ) α + m + 1ξm,n
hFα (f )/ξm,n iγα+1 = p
(µ, λ)
2(α + 1)
Γ+
√
α
− m + 1ξm+1,n
(µ, λ) dγα (µ, λ)
s
α+m+1
hFα (f )/(i)2m+n Fα (eαm,n )iγα
=
2(α + 1)
s
m+1
−
hFα (f )/(i)2m+2+n Fα (eαm+1,n )iγα ,
2(α + 1)
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
hence, according to the Parseval’s equality (2.11), we obtain
s
α+m+1
α+1
(−i)2m+n hf /eαm,n iνα
hFα (f )/ξm,n
iγα+1 =
2(α + 1)
s
m+1
+
(−i)2m+n hf /eαm+1,n iνα .
2(α + 1)
Contents
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4.
Heisenberg-Pauli-Weyl Inequality for the Fourier Transform
Fα
In this section, we will prove the main result of this work, that is the HeisenbergPauli-Weyl inequality for the Fourier transform Fα connected with the RiemannLiouville operator Rα . Next we give a generalization of this result, for this we need
the following important lemma.
Lemma 4.1. Let f ∈ L2 (dνα ), such that
k|(r, x)|f k2,να < +∞ and
S. Omri and L.T. Rachdi
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞,
then
1
2
(4.1) k|(r, x)|f k22,να +k(µ2 +2λ2 ) Fα (f )k22,γα =
+∞
X
Title Page
(m, n) ∈ N2 .
Proof. Let f ∈ L2 (dνα ), such that
∀(r, x) ∈ R+ × R;
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Page 21 of 45
f (r, x) =
+∞
X
am,n eαm,n (r, x),
m,n=0
k|(r, x)|f k2,να < +∞ and
1
2
k(µ + 2λ ) Fα (f )k2,γα < +∞,
2
2
then the functions (r, x) 7−→ rf (r, x) and (r, x) 7−→ xf (r, x) belong to the space
L2 (dνα ), in particular f ∈ L2 (dνα ) ∩ L2 (dνα+1 ). In the same manner, the functions
(µ, λ) 7−→ (µ2 + λ2 ) 2 Fα (f )(µ, λ),
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and assume that
1
vol. 9, iss. 3, art. 88, 2008
2α+4m+2n+3 |am,n |2 ,
m,n=0
where am,n = hf /eαm,n iνα ;
Heisenberg-Pauli-Weyl
Uncertainty Principle
and (µ, λ) 7−→ λFα (f )(µ, λ)
Close
belong to the space L2 (dγα ). In particular, by the relation (2.7), we deduce that
Fα (f ) ∈ L2 (dγα ) ∩ L2 (dγα+1 ), and we have
Z +∞ Z
2
krf k2,να =
r2 |f (r, x)|2 dνα (r, x)
0
R
= 2(α + 1)kf k22,να+1
+∞
X
2
hf /eα+1
= 2(α + 1)
m,n iνα+1 ,
m,n=0
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
hence, according to the relation (3.6), we obtain
(4.2)
krf k22,να
+∞
X
√
√
α + m + 1am,n − m + 1am+1,n 2 .
=
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Contents
m,n=0
Similarly, we have
kxf k22,να =
Z
+∞ Z
0
x2 |f (r, x)|2 dνα (r, x)
m,n=0
and by the relation (3.1), we get
(4.3)
kxf k22,να
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Page 22 of 45
R
+∞
+∞
X
X
2
α
hf /xeαm,n iνα 2 ,
=
hxf /em,n iνα =
m,n=0
JJ
2
r
r
n+1
n
=
am,n+1 +
am,n−1 .
2
2
m,n=0
+∞
X
By the same arguments, and using the relations (3.2), (3.7) and the Parseval’s equality (2.11), we obtain
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(4.4)
1
k(µ2 + λ2 ) 2 Fα (f )k22,γα =
+∞
X
√
√
α + m + 1am,n + m + 1am+1,n 2 ,
m,n=0
and
(4.5)
kλFα (f )k22,γα
2
r
+∞ r
X
n
n+1
=
am,n+1 −
am,n−1 .
2
2
m,n=0
Combining now the relations (4.2), (4.3), (4.4) and (4.5), we deduce that
1
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
k|(r, x)|f k22,να + k(µ2 + 2λ2 ) 2 Fα (f )k22,γα
1
= krf k22,να + k(µ2 + λ2 ) 2 Fα (f )k22,γα + kxf k22,να + kλFα (f )k22,γα
+∞ X
=2
(α + m + 1)|am,n |2 + (m + 1)|am+1,n |2
m,n=0
+∞ X
n+1
n
2
2
+2
|am,n+1 | + |am,n−1 |
2
2
m,n=0
=2
+∞
X
(α + m + 1)|am,n |2 + 2
m,n=0
+∞
X
=
m,n=0
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Page 23 of 45
m|am,n |2
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m,n=0
+∞
+∞
X
X
n
n+1
2
+2
|am,n | + 2
|am,n |2
2
2
m,n=0
m,n=0
+∞
X
Title Page
2α + 4m + 2n + 3 |am,n |2 .
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Close
Remark 1. From the relation (4.1), we deduce that for all f ∈ L2 (dνα ), we have
(4.6)
1
k|(r, x)|f k22,να + k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ≥ (2α + 3)kf k22,να ,
with equality if and only if
∀(r, x) ∈ R+ × R;
f (r, x) = Ce−
r 2 +x2
2
;
C ∈ C.
Lemma 4.2. Let f ∈ L2 (dνα ) such that,
S. Omri and L.T. Rachdi
1
2
k(µ2 + 2λ2 ) Fα (f )k2,γα < +∞,
k|(r, x)|f k2,να < +∞ and
then
1) For all t > 0,
vol. 9, iss. 3, art. 88, 2008
Title Page
1
1
k|(r, x)|f k22,να + t2 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ≥ (2α + 3)kf k22,να .
2
t
2) The following assertions are equivalent
1
2
i) k|(r, x)|f k2,να k(µ2 + 2λ2 ) Fα (f )k2,γα
ii) There exists t0 > 0, such that
2α + 3
=
kf k22,να .
2
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Page 24 of 45
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1
2
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να ,
Proof. 1) Let f ∈ L2 (dνα ) satisfy the hypothesis.For all t > 0 we put ft (r, x) =
f (tr, tx), and then by a simple change of variables, we get
kft k22,να =
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where ft0 (r, x) = f (t0 r, t0 x).
(4.7)
Heisenberg-Pauli-Weyl
Uncertainty Principle
1
t2α+3
kf k22,να ,
and
k|(r, x)|ft k22,να =
(4.8)
1
t2α+5
k|(r, x)|f k22,να .
For all (µ, λ) ∈ Γ,
Fα (ft )(µ, λ) =
(4.9)
1
t2α+3
Fα (f )
µ λ
,
t t
,
Heisenberg-Pauli-Weyl
Uncertainty Principle
and by using the relation (2.6), we deduce that
(4.10)
1
k(µ2 + 2λ2 ) 2 Fα (ft )k22,γα =
1
t2α+1
S. Omri and L.T. Rachdi
1
k(µ2 + 2λ2 ) 2 Fα (f )k22,γα .
Then, the desired result follows by replacing f by ft in the relation (4.6).
vol. 9, iss. 3, art. 88, 2008
Title Page
Contents
2) Let f ∈ L2 (dνα ); f 6= 0.
• Assume that
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
2α + 3
kf k22,να .
2
2
1
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1
2
By Theorem 2.3, we have k(µ + 2λ ) Fα (f )k2,γα 6= 0, then for
s
k|(r, x)|f k2,να
t0 =
,
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
we have
1
1
k|(r, x)|f k22,να + t20 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα = (2α + 3)kf k22,να ,
2
t0
and this is equivalent to
2
JJ
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να .
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• Conversely, suppose that there exists t1 > 0, such that
1
k|(r, x)|ft1 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft1 )k22,γα = (2α + 3)kft1 k22,να .
This is equivalent to
1
1
(4.11)
k|(r, x)|f k22,να + t21 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα = (2α + 3)kf k22,να .
2
t1
However, let h be the function defined on ]0, +∞[, by
1
1
h(t) = 2 k|(r, x)|f k22,να + t2 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ,
t
then, the minimum of the function h is attained at the point
s
k|(r, x)|f k2,να
t0 =
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
and
1
h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα .
Thus by 1) of this lemma, we have
1
h(t1 ) ≥ h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα ≥ (2α + 3)kf k22,να .
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
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According to the relation (4.11), we deduce that
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1
h(t1 ) = h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα = (2α + 3)kf k22,να .
Theorem 4.3 (Heisenbeg-Pauli-Weyl inequality). For all f ∈ L2 (dνα ), we have
(4.12)
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα ≥
(2α + 3)
kf k22,να
2
Close
with equality if and only if
∀(r, x) ∈ R+ × R;
f (r, x) = Ce
−r
2 +x2
2t2
0
t0 > 0, C ∈ C.
;
1
Proof. It is obvious that if f = 0, or if k|(r, x)|f k2,να = +∞, or k(µ2 +2λ2 ) 2 Fα (f )k2,γα
= +∞, then the inequality (4.12) holds.
1
Let us suppose that k|(r, x)|f k2,να + k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞, and f 6= 0.
By 1) of Lemma 4.2, we have for all t > 0
1
2
2
2
2 12
k|(r,
x)|f
k
+
t
k(µ
+
2λ
) Fα (f )k22,γα ≥ (2α + 3)kf k22,να ,
2,ν
α
t2
and the result follows if we pick
s
k|(r, x)|f k2,να
t=
.
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
By 2) of Lemma 4.2, we have
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
2α + 3
kf k22,να ,
2
if and only if there exists t0 , such that
1
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να ,
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
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and according to Remark 1, this is equivalent to
ft0 (r, x) = Ce
−r
which means that
f (r, x) = Ce
−r
2 +x2
2
2 +x2
2t2
0
Close
;
C ∈ C,
;
C ∈ C.
The following gives a generalization of the Heisenberg-Pauli-Weyl inequality.
Theorem 4.4. Let a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, then for all
f ∈ L2 (dνα ) we have
aη
2α + 3
η
1−η
a
2
2 2b
k|(r, x)| f k2,να k(µ + 2λ ) Fα (f )k2,γα ≥
kf k2,να
2
Heisenberg-Pauli-Weyl
Uncertainty Principle
with equality if and only if a = b = 1 and
∀(r, x) ∈ R+ × R;
f (r, x) = Ce
−r
2 +x2
2t2
0
S. Omri and L.T. Rachdi
;
t0 > 0, C ∈ C.
Proof. Let f ∈ L2 (dνα ), f 6= 0, such that
vol. 9, iss. 3, art. 88, 2008
Title Page
b
2
k|(r, x)| f k2,να + k(µ + 2λ ) Fα (f )k2,γα < +∞.
a
2
2
Contents
Then for all a > 1, we have
1
a
1
a0
2
a
1
2
2
a0
1
2
a0 ,ν
k|(r, x)|a f k2,να kf k2,να = k|(r, x)|2 |f | ka,να k|f | k
where a0 is defined as usual by a0 =
a
.
a−1
α
,
By Hölder’s inequality we get
1
1
a
a0
k|(r, x)|a f k2,ν
kf k2,ν
> k|(r, x)|f k2,να .
α
α
The strict inequality here is justified by the fact that if f 6= 0, then the functions
|(r, x)|2a |f |2 and |f |2 cannot be proportional.Thus for all a ≥ 1, we have
1
(4.13)
a
k|(r, x)|a f k2,ν
≥
α
with equality if and only if a = 1.
k|(r, x)|f k2,να
1
a0
kf k2,να
.
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In the same manner and using Plancherel’s Theorem 2.4, we have for all b ≥ 1
(4.14)
b
2
1
1
b
k(µ + 2λ ) Fα (f )k2,γα ≥
2
2
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
1
b0
kFα (f )k2,γ
α
1
≥
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
1
b0
.
kf k2,να
with equality if and only if b = 1.
b
Let η = a+b
, then by the relations (4.13), (4.14) and for all a, b ≥ 1, we have
ηa

2
2 12
b
2λ ) Fα (f )k2,γα 
 k|(r, x)|f k2,να k(µ +
k|(r, x)|a f kη2,να k(µ2 +2λ2 ) 2 Fα (f )k1−η
,
1
1
2,γα ≥
+
a0
b0
kf k2,ν
α
b
k|(r, x)|a f kη2,να k(µ2 + 2λ2 ) 2 Fα (f )k1−η
2,γα ≥
2α + 3
2
ηa
kf k2,να ,
with equality if and only if a = b = 1 and
f (r, x) = Ce
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
Contents
with equality if and only if a = b = 1.
Applying Theorem 4.3, we obtain
∀(r, x) ∈ R+ × R;
Heisenberg-Pauli-Weyl
Uncertainty Principle
−r
2 +x2
2t2
0
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;
t0 > 0, C ∈ C.
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Remark 2. In the particular case when a = b = 2, the previous result gives us the
Heisenberg-Pauli-Weyl inequality for the fourth moment of Heisenberg
2
2α + 3
2
2
2
k|(r, x)| f k2,να k(µ + 2λ )Fα (f )k2,γα >
kf k22,να .
2
5.
The Local Uncertainty Principle
Theorem 5.1. Let ξ be a real number such that 0 < ξ < 2α+3
, then for all f ∈
2
2
L (dνα ), f 6= 0, and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we
have
ZZ
2ξ
(5.1)
|Fα (f )(µ, λ)|2 dγα (µ, λ) < Kα,ξ γα (E) 2α+3 k|(r, x)|ξ f k22,να ,
E
where
Kα,ξ =
2ξ
! 2α+3
2α + 3 − 2ξ
5
ξ 2 2α+ 2 Γ
α+
3
2
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
2α + 3
2α + 3 − 2ξ
2
.
Proof. For all s > 0, we put
Bs = (r, x) ∈ R+ × R ; r2 + x2 < s2 .
Let f ∈ L2 (dνα ). By Minkowski’s inequality, we have
12
Z Z
(5.2)
|Fα (f )(µ, λ)|2 dγα (µ, λ)
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vol. 9, iss. 3, art. 88, 2008
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= kFα (f )1E k2,γα
≤ kFα (f 1Bs )1E k2,γα + kFα (f 1Bsc )1E k2,γα
1
≤ γα (E) 2 kFα (f 1Bs )k∞,γα + kFα (f 1Bsc )k2,γα .
Applying the relation (2.10), we deduce that for every s > 0, we have
Z Z
21
2
|Fα (f )(µ, λ)| dγα (µ, λ)
(5.3)
E
1
≤ γα (E) 2 kf 1Bs k1,να + kFα (f 1Bsc )k2,γα .
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On the other hand, by Hölder’s inequality we have
(5.4)
kf 1Bs k1,να ≤ k|(r, x)|ξ f k2,να k|(r, x)|−ξ 1Bs k2,να
s
ξ
= k|(r, x)| f (r, x)k2,να (5.5)
α+ 12
2
Γ α+
2α+3−2ξ
2
3
2
12 .
(2α + 3 − 2ξ)
Heisenberg-Pauli-Weyl
Uncertainty Principle
By Plancherel’s theorem 2.4, we have also
S. Omri and L.T. Rachdi
(5.6)
kFα (f 1 )k2,γα = kf 1 k2,να
Bsc
Bsc
ξ
vol. 9, iss. 3, art. 88, 2008
−ξ
≤ k|(r, x)| f k2,να k|(r, x)| 1Bsc k∞,να
= s−ξ k|(r, x)|ξ f k2,να .
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Contents
Combining the relations (5.3), (5.5) and (5.6), we deduce that for all s > 0 we have
Z Z
(5.7)
12
|Fα (f )(µ, λ)| dγα (µ, λ)
≤ gα,ξ (s)k|(r, x)|ξ f (r, x)k2,να ,
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where gα,ξ is the function defined on ]0, +∞[ by
gα,ξ (s) = s−ξ +
γα (E)
1
2α+ 2 Γ α + 32 (2α + 3 − 2ξ)
Thus, the inequality (5.7) holds for
s0 =
! 1
5
ξ 2 2α+ 2 Γ( α + 32 ) 2α+3
,
γα (E)(2α + 3 − 2ξ)
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! 12
s
2α+3−2ξ
2
.
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however
1
ξ
2
gα,ξ (s0 ) = γα (E) 2α+3 Kα,ξ
,
where
Kα,ξ =
2α + 3 − 2ξ
5
ξ 2 2α+ 2 Γ(α + 32 )
2ξ
! 2α+3
2α + 3
2α + 3 − 2ξ
2
.
Let us prove that the equality in (5.1) cannot hold. Indeed, suppose that
Z Z
2ξ
|Fα (f )(µ, λ)|2 dγα (µ, λ) = Kα,ξ γα (E) 2α+3 k|(r, x)|ξ f k22,να .
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
E
Then
Title Page
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ) = gα,ξ (s0 )2 k|(r, x)|ξ f k22,να ,
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E
and therefore by the relations (5.2), (5.3) and (5.4), we get
(5.8)
kFα (f 1Bs0 )1E k2,γα
1
= γα (E) 2 kFα (f 1Bs0 )k∞,γα ,
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(5.9)
kf 1Bs0 k1,να = kFα (f 1Bs0 )k∞,γα ,
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and
(5.10)
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kf 1Bs0 k1,να = k|(r, x)ξ f k2,να k|(r, x)−ξ 1Bs0 k2,να .
However, if f satisfies the equality (5.10), then there exists C > 0, such that
|(r, x)|2ξ f (r, x)2 = C|(r, x)|−2ξ 1Bs0 ,
Close
hence
∀(r, x) ∈ R+ × R;
(5.11)
f (r, x) = CeiΦ(r,x) |(r, x)|−2ξ 1Bs0 ,
where Φ is a real measurable function on R+ × R.
But if f satisfies the relation (5.9), then there exists (µ0 , λ0 ) ∈ Γ+ , such that
kf k1,να = kFα (f )k∞,γα = Fα (f )(µ0 , λ0 ).
So, there exists θ0 ∈ R satisfying
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
Fα (f )(µ0 , λ0 ) = e kf k1,να ,
vol. 9, iss. 3, art. 88, 2008
and therefore
q
Z +∞ Z
−2ξ
iΦ(r,x)−iλ0 x−iθ0
iθ0
2
2
Ce
|(r, x)| 1Bs0 (r, x) e
jα r µ0 + λ0 − 1 dνα (r, x) = 0.
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iθ0
0
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R
This implies that for almost every (r, x) ∈ R+ × R,
q
iΦ(r,x)−iλ0 x−iθ0
e
jα r µ20 + λ20 = 1.
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Hence, we deduce that for all r ∈ R+ ,
q
jα r µ20 + λ20 = 1.
Using the relation (2.2), it follows that µ20 + λ20 = 0, or µ0 = i|λ0 |, and then
eiΦ(r,x) = eiθ0 +iλ0 x .
Replacing in (5.11), we get
f (r, x) = C 0iλ0 x |(r, x)|−2ξ 1Bs0 (r, x).
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Now, the relation (5.8) means that for almost every (µ, λ) ∈ E, we have
Fα (f )(µ, λ) = kFα (f )k∞,γα = Fα (f )(µ0 , λ0 ),
which implies that for almost every (µ, λ) ∈ E, we have
Fα (f )(µ, λ) = eiψ(µ,λ) Fα (f )(µ0 , λ0 ),
where ψ is a real measurable function on E, and therefore
Z +∞ Z
p
0iψ(µ,λ)
C
|(r, x)|−2ξ 1Bs0 (r, x) e−iλx+iλ0 x−iψ(µ,λ) jα r µ2 + λ2 − 1 dνα (r, x) = 0.
0
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
R
Consequently for all (r, x) ∈ R+ × R,
−iλx+iλ0 x−iψ(µ,λ)
e
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p
2
2
jα r µ + λ = 1,
Contents
which implies that λ = λ0 and µ = µ0 .
However, since γα (E) > 0, this contradicts the fact that for almost every (µ, λ) ∈ E,
Fα (f )(µ, λ) = Fα (f )(µ0 , λ0 ),
and shows that the inequality in (5.1) is strictly satisfied.
Lemma 5.2. Let ξ be a real number such that ξ >
function f on R+ × R we have
kf k21,να
(5.12)
≤
(2α+3)
ξ
2−
Mα,ξ kf k2,να
then for all measurable
(2α+3)
ξ
k|(r, x)| f k2,να ,
where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
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2α+3
,
2
ξ
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2α+3
2ξ
.
2α+3
sin π 2ξ
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with equality in (5.12) if and only if there exist a, b > 0 such that
|f (r, x)| = (a + b|(r, x)|2ξ )−1 .
Proof. We suppose naturally that f 6= 0. It is obvious that the inequality (5.12) holds
if kf k2,να = +∞ or k|(r, x)|ξ f k2,να = +∞.
Assume that kf k2,να + k|(r, x)|ξ f k2,να < +∞.From the hypothesis 2ξ > 2α + 3,
we deduce that for all a, b > 0, the function
(r, x) 7−→ (a + b|(r, x)|2ξ )−1
S. Omri and L.T. Rachdi
belongs to L1 (dνα ) ∩ L2 (dνα ) and by Hölder’s inequality, we have
2
1 2
2ξ − 12 (5.13)
(1
+
|(r,
x)|
)
kf k21,να ≤ (1 + |(r, x)|2ξ ) 2 f 2,να
2,να
2
2
ξ
2
2ξ − 12 ≤ kf k2,να + k|(r, x)| f k2,να (1 + |(r, x)| ) vol. 9, iss. 3, art. 88, 2008
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However, by standard calculus, we have
2
π
2ξ − 21 .
(1
+
|(r,
x)|
)
=
3
2α+3
α+ 32
2,να
ξ2
Γ α + 2 sin π 2ξ
Thus
(5.14)
π
2
ξ
2
≤
kf
k
+
k|(r,
x)|
f
k
,
2,ν
2,ν
α
α
3
2α+3
α+ 23
ξ2
Γ α + 2 sin π 2ξ
with equality in (5.14) if and only if we have equality in (5.13), that is there exists
C > 0 satisfying
1
1
(1 + |(r, x)|2ξ ) 2 |f (r, x)| = C(1 + |(r, x)|2ξ )− 2 ,
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.
2,να
kf k21,να
Heisenberg-Pauli-Weyl
Uncertainty Principle
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or
|f (r, x)| = C(1 + |(r, x)|2ξ )−1 .
(5.15)
For t > 0, we put as above, ft (r, x) = f (tr, tx), then we have
kft k21,να =
(5.16)
1
t4α+6
kf k21,να ,
Heisenberg-Pauli-Weyl
Uncertainty Principle
and
S. Omri and L.T. Rachdi
1
vol. 9, iss. 3, art. 88, 2008
Replacing f by ft in the relation (5.14), we deduce that for all t > 0, we have
π
2α+3
2
2α+3−2ξ
ξ
2
kf k21,να ≤
t
kf
k
+t
k|(r,
x)|
f
k
2,να
2,να .
3
ξ2α+ 2 Γ α + 32 sin π 2α+3
2ξ
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In particular, for
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k|(r, x)|ξ ft k22,να =
(5.17)
t = t0 =
k|(r, x)|ξ f k22,να .
2ξ+2α+3
t
(2ξ − 2α − 3)k|(r, x)|ξ f k22,να
(2α + 3)kf k22,να
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! 2ξ1
,
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we get
2−
(2α+3)
ξ
kf k21,να ≤ Mα,ξ kf k2,να
(2α+3)
k|(r, x)|ξ f k2,ναξ ,
where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
sin π 2α+3
2ξ
Close
Now suppose that we have equality in the last inequality. Then we have equality in
(5.14) for ft0 and by means of (5.15), we obtain
|ft0 (r, x)| = C(1 + |(r, x)|2ξ )−1 ,
and then
|f (r, x)| = (a + b|(r, x)|2ξ )−1 .
Heisenberg-Pauli-Weyl
Uncertainty Principle
Theorem 5.3. Let ξ be a real number such that ξ > 2α+3
. Then for all f ∈ L2 (dνα ),
2
f 6= 0, and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we have
Z Z
2α+3
2− 2α+3
(5.18)
|Fα (f )(µ, λ)|2 dγα (µ, λ) < Mα,ξ γα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα ,
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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E
Contents
where
Mα,ξ =
π
1
2α+ 2 Γ α +
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
2α+3
sin π 2ξ
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Moreover, Mα,ξ is the best (the smallest) constant satisfying (5.18).
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Proof.
• Suppose that the right-hand side of (5.18) is finite. Then, according to
Lemma 5.2, the function f belongs to L1 (dνα ) and we have
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ) ≤ γα (E)kFα (f )k2∞,γα
E
≤ γα (E)kf k21,να
2−
(2α+3)
ξ
≤ γα (E)Mα,ξ kf k2,να
(2α+3)
k|(r, x)|ξ f k2,ναξ ,
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where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
2α+3
sin π 2ξ
• Let us prove that the equality in (5.18) cannot hold.
Indeed, suppose that
Z Z
2α+3
2− 2α+3
|Fα (f )(µ, λ)|2 dγα (µ, λ) = Mα,ξ γα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα .
E
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Then
Z Z
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|Fα (f )(µ, λ)|2 dγα (µ, λ) = γα (E)kFα (f )k2∞,γα ,
(5.19)
Contents
Γ+
(5.20)
kFα (f )k∞,γα = kf k1,να ,
and
(5.21)
2− 2α+3
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kf k21,να = Mα,ξ kf k2,να ξ k|(r, x)|ξ f k2,να .
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∀(r, x) ∈ R+ × R;
f (r, x) = ϕ(r, x)(a + b|(r, x)|2ξ )−1 ,
with |ϕ(r, x)| = 1; a, b > 0.
On the other hand, there exists (µ0 , λ0 ) ∈ Γ+ such that
(5.23)
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(2α+3)
ξ
Applying Lemma 5.2 and the relation (5.21), we deduce that
(5.22)
JJ
kFα (f )k∞,γα = |Fα (f )(µ0 , λ0 )| = eiθ0 Fα (f )(µ0 , λ0 );
θ0 ∈ R.
Close
Combining now the relations (5.20), (5.22) and (5.23), we get
q
Z +∞ Z iθ0
−iλ0 x
2
2
1 − e ϕ(r, x)jα r µ0 + λ0 e
(a + b|(r, x)|2ξ )−1 dνα (r, x) = 0.
0
R
This implies that for almost every (r, x) ∈ R+ × R,
q
−iλ0 x iθ0
2
2
e
e ϕ(r, x)jα r µ0 + λ0 = 1.
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Since |ϕ(r, x)| = 1, we deduce that for all r ∈ R+ ,
q
jα r µ20 + λ20 = 1.
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Using the relation (2.2), it follows that µ20 + λ20 = 0, or µ0 = i|λ0 |, and therefore
ϕ(r, x) = e−iθ0 eiλ0 x .
Replacing in (5.22), we get
f (r, x) = Ce
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iλ0 x
2ξ −1
(a + b|(r, x)| ) ;
|C| = 1.
Now, the relation (5.19) means that
Z Z
kFα (f )k2∞,γα − |Fα (f )(µ, λ)|2 dγα (µ, λ) = 0.
E
Hence, for almost every (µ, λ) ∈ E,
(5.24)
Contents
|Fα (f )(µ, λ)| = kFα (f )k∞,γα = eiθ0 Fα (f )(µ0 , λ0 ).
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Let Ψ(µ, λ) ∈ R, such that
|Fα (f )(µ, λ)| = eiΨ(µ,λ) Fα (f )(µ, λ).
Then from (5.24), for almost every (µ, λ) ∈ E,
eiΨ(µ,λ) Fα (f )(µ, λ) = eiθ0 Fα (f )(µ0 , λ0 ),
and therefore
Z +∞ Z
h
p
i
i(λ0 −λ)x iΨ(µ,λ)
2ξ −1
2
2
(a + b|(r, x)| )
1−e
e
jα r µ + λ
dνα (r, x) = 0.
0
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
R
Consequently, for all (r, x) ∈ R+ × R,
Title Page
p
ei(λ0 −λ)x eiΨ(µ,λ) jα r µ2 + λ2 = 1,
which implies that λ = λ0 and µ = µ0 .
However, since γα (E) > 0, this contradicts the fact that for almost every (µ, λ) ∈ E,
eiΨ(µ,λ) Fα (f )(µ, λ) = eiθ0 Fα (f )(µ0 , λ0 ),
and shows that the inequality in (5.18) is strictly satisfied.
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• Let us prove that the constant Mα,ξ is the best one satisfying (5.18).
Let A be a positive constant, such that for all f ∈ L2 (dνα ),
Z Z
2α+3
2− 2α+3
(5.25)
|Fα (f )(µ, λ)|2 dγα (µ, λ) ≤ Aγα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα .
E
Close
We assume that k|(r, x)|ξ f k2,να < +∞. Then by Lemma 5.2, f belongs to L1 (dνα ).
Replacing f by ft (r, x) = f (tr, tx) in (5.25), and using the relations (4.7), (4.9) and
(5.17), we deduce that for all t > 0
2
Z Z 2α+3
2− 2α+3
µ
λ
ξ
ξ
ξ
Fα (f )
dγα (µ, λ) ≤ Akf k2,ν
,
k|(r,
x)|
f
k
.
2,ν
α
α
t t
E
Using the dominate convergence theorem and the relations (2.1), (2.3), (2.9) and
(2.10), we deduce that
2
Z Z µ λ 2
lim
Fα (f ) t , t dγα (µ, λ) = |Fα (f )(0, 0)| γα (E).
t→+∞
E
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
Title Page
Consequently, for all f ∈ L2 (dνα ), such that k|(r, x)|ξ f k2,να < +∞,
|Fα (f )(0, 0)|2 ≤
(5.26)
2α+3
2− 2α+3
Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
Now, let f ∈ L2 (dνα ), such that k|(r, x)|ξ f k2,να < +∞, and let (µ, λ) ∈ Γ. Putting
p
2
2
g(r, x) = jα r µ + λ e−iλx f (r, x),
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then g ∈ L (dνα ), and k|(r, x)| gk2,να < +∞. Moreover, Fα (g)(0, 0) = Fα (f )(µ, λ),
and by (5.26), it follows that
2
ξ
2
|Fα (f )(µ, λ)| ≤
2α+3
2− 2α+3
Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
Thus, for all f ∈ L2 (dνα ) such that k|(r, x)|ξ f k2,να < +∞, we have
(5.27)
2− 2α+3
2α+3
kFα (f )k2∞,γα ≤ Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
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Taking f0 (r, x) = (1 + |(r, x)|2ξ )−1 , we have

kFα (f0 )k2∞,γα = kf0 k21,να = 
kf0 k22,να =
ξ
k|(r, x)|
α+ 52
ξ22
f0 k22,να
α+ 32
ξ2
2
π
 ,
3
2α+3
Γ α + 2 sin π 2ξ
(2ξ − 2α − 3)π
,
Γ α + 32 sin π 2α+3
2ξ
(2α + 3)π
.
=
5
ξ 2 2α+ 2 Γ α + 32 sin π 2α+3
2ξ
Replacing f by f0 in the relation (5.27), we obtain A ≥ Mα,ξ .
This completes the proof of Theorem 5.3.
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
vol. 9, iss. 3, art. 88, 2008
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References
[1] G. ANDREWS, R. ASKEY AND R. ROY, Special Functions. Cambridge University Press, 1999.
[2] C. BACCAR, N.B. HAMADI AND L.T. RACHDI, Inversion formulas for the
Riemann-Liouville transform and its dual associated with singular partial differential operators, International Journal of Mathematics and Mathematical
Sciences, 2006 (2006), Article ID 86238, 1-26.
Heisenberg-Pauli-Weyl
Uncertainty Principle
S. Omri and L.T. Rachdi
[3] G. BATTLE, Heisenberg inequalities for wavelet states, Appl. and Comp. Harmonic Anal., 4(2) (1997), 119–146.
vol. 9, iss. 3, art. 88, 2008
[4] M.G. COWLING AND J.F. PRICE, Generalizations of Heisenberg’s inequality
in Harmonic analysis (Cortona, 1982), Lecture Notes in Math., 992, Springer,
Berlin (1983), 443–449. MR0729369 (86g:42002b).
Title Page
[5] N.G. DE BRUIJN, Uncertainty principles in Fourier analysis, Inequalities
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Press, New York, (1967), 57–71.
[6] A. ERDELY, Asymptotic Expansions. Dover Publications, Inc.New-York,
1956.
[7] W.G. FARIS, Inequalities and uncertainty principles, J. Math. Phys., 19 (1978),
461–466.
[8] G.B. FOLLAND AND A. SITARAM, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207–238.
[9] G.H. HARDY, A theorem concerning Fourier transform, J. London Math. Soc.,
8 (1933), 227–231.
Contents
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[10] 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,
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[11] N.N. LEBEDEV, Special Functions and their Applications, Dover Publications
Inc., New-York, 1972.
[12] W. MAGNUS, F. OBERHETTINGER AND F.G. TRICOMI, Tables of Integral
Transforms, McGraw-Hill Book Compagny, Inc. New York/London/Toronto,
1954.
[13] G.W. MORGAN, A note on Fourier transforms, J. London. Math. Soc., 9
(1934), 178–192.
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