Hardy’s and Miyachi’s theorems for the first Hankel – Clifford transform
UDC 517.5 We present an analog of Hardy's and Miyachi's theorems for the first Hankel$\,--\,$,Clifford transform.
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| author | El, Kassimi M. Fahlaoui, S. Ель, Кассимі М. Фахлауі, С. |
| author_facet | El, Kassimi M. Fahlaoui, S. Ель, Кассимі М. Фахлауі, С. |
| author_sort | El, Kassimi M. |
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| description | UDC 517.5
We present an analog of Hardy's and Miyachi's theorems for the first Hankel$\,--\,$,Clifford transform. |
| first_indexed | 2026-03-24T02:06:17Z |
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UDC 517.5
M. El Kassimi, S. Fahlaoui (Dep. Math. and Comput. Sci., Univ. Moulay Ismaı̈l, Meknès, Morocco)
HARDY’S AND MIYACHI’S THEOREMS
FOR THE FIRST HANKEL – CLIFFORD TRANSFORM
ТЕОРЕМИ ГАРДI ТА МIЯЧI
ПРО ПЕРШЕ ПЕРЕТВОРЕННЯ ГАНКЕЛЯ – КЛIФФОРДА
We present an analog of Hardy’s and Miyachi’s theorems for the first Hankel – Clifford transform.
Наведено аналог теорем Гардi та Мiячi про перше перетворення Ганкеля – Клiффорда.
1. Introduction. In signal processing, the uncertainty principle states that the signal variances
product in the time and frequency domains has a lower bound. The mathematical formulation of this
fact is that the uncertainty principles for the Fourier transform relate the variances of a function and
its Fourier transform which cannot both be simultaneously sharped localized. One example of this is
the Heisenberg uncertainty principle concerning the position and the momentum wave functions in
quantum physics. Many mathematical formulations of this general fact can be found in [7]. Namely
theorems of Hardy [6], Cowling and Price [3] and Miyachi [11]. In 1933, Hardy [6] demonstrated
the following theorem: if | f(x)| \leq Ce - ax2
and | \widehat f(y)| \leq Ce - by2 for some positive numbers a, b
and C, then f = 0 whenever ab > 1/4. If ab = 1/4, then the function f is a constant multiple of
e - ax2
, and if ab < 1/4, then are infinitely functions which realise both conditions. In 1997, Miyachi
[11] proved the next theorem:
Theorem 1.1. Let f be an integrable function on \BbbR such that
eax
2
f \in L1(\BbbR ) + L\infty (\BbbR ).
Further, assume that \int
\BbbR
\mathrm{l}\mathrm{o}\mathrm{g}+
\Biggl(
| eb\lambda 2 \widehat f(\lambda )|
c
\Biggr)
d\lambda < \infty
for some positive numbers a, b and c. If ab = 1/4, then f is a constant multiple of the Gaussian
e - ax2
.
The first Hankel – Clifford transform has great importance in solving problems involving cylin-
drical boundaries. The Hankel – Clifford transformation is useful mathematical tools in solving a
certain class of partial differential equations, involving the generalized Kepinsky – Myller – Lebedev
differential operator.
In [4] the authors gave a version of Morgan and Cowling – Price in the case of the first Hankel –
Clifford transform.
The purpose of this paper is to demonstrate the Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform.
First, we would like to mention some main results of the first Hankel – Clifford operator. In the
second, we are going to review two principal lemmas of the complex variables theory, which are a
c\bigcirc M. EL KASSIMI, S. FAHLAOUI, 2019
710 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
HARDY’S AND MIYACHI’S THEOREMS FOR THE FIRST HANKEL – CLIFFORD TRANSFORM 711
version of the Phragmen – Lindelöff theorem. After that, we are going to give a version of Hardy’s
theorem associated with the first Hankel – Clifford transform. In the last section, we are proving an
analog of Miyachi’s theorem for the first Hankel – Clifford transform.
In [2, 9, 12], the first Hankel – Clifford transform of order \mu \geq 0 was introduced by
(h1,\mu \varphi )(\lambda ) = \lambda \mu
+\infty \int
0
C\mu (x\lambda )\varphi (x) dx,
where C\mu is the Bessel – Clifford function of the first kind of order \mu [5] and defined by
C\mu (x) =
\infty \sum
k=0
( - 1)kxk
k! \Gamma (\mu + k + 1)
which is a solution of the differential equation
x
\partial 2
\partial x2
y + (y + 1)
\partial
\partial x
y + y = 0
and is closely related with the Bessel function of the first kind J\mu and index \mu by
C\mu (x) = x
- \mu
2 J\mu
\bigl(
2x
1
2
\bigr)
,
where Bessel function J\mu is defined in [8] by
2\mu x - \mu J\mu (x) =
\infty \sum
n=0
( - 1)n
n! \Gamma (\mu + n+ 1)
\Bigl( x
2
\Bigr) 2n
.
The inversion formula of the first Hankel – Clifford transform is defined by
\varphi (x) = (h - 1
1,\mu h1,\mu \varphi )(x) = x\mu
+\infty \int
0
C\mu (\lambda , x)\varphi (\lambda ) d\lambda .
From [10], the first Hankel – Clifford transform satisfied that
h - 1
1,\mu = h1,\mu for \mu \geq 0.
The demonstration of the basic outcomes relies on the following two complex variable lemmas,
which will be presented in this section.
Lemma 1.1. Let h be an entire function on \BbbC such that
| h(z)| \leq Cea| z|
2 \forall z \in \BbbC
and
| h(t)| \leq Ae - at2 \forall t \in \BbbR
for some positive constants a, B and C. Then h(z) = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t} e - az2 , z \in \BbbC .
Proof. See [13] (Lemma 2.1).
Let us define \mathrm{l}\mathrm{o}\mathrm{g}+(x) = \mathrm{l}\mathrm{o}\mathrm{g}(x) if x > 1, and \mathrm{l}\mathrm{o}\mathrm{g}+(x) = 0 otherwise. We also need the
following lemma.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
712 M. EL KASSIMI, S. FAHLAOUI
Lemma 1.2. Let h an entire function and suppose that, there exist constants A, B > 0 such
that
| h(z)| \leq AeB\Re (z)2 \forall z \in \BbbC
and
+\infty \int
- \infty
\mathrm{l}\mathrm{o}\mathrm{g}+ | h(t)| dt < \infty .
Then h is a constant function.
Proof. See [11] (Lemma 4).
We need also an estimated results for the Bessel’s function, we have the next lemma [1].
Lemma 1.3. Let \mu - 1/2. We have the following results:
(1) | j\mu (x)| \leq 1,
(2) 1 - j\mu (x) = O(1), x \geq 1,
(3) 1 - j\mu (x) = O(x2), 0 \leq x \leq 1,
(4)
\surd
hxJ\mu (hx) = O(1), hx \geq 0.
Since the last formula and the definition of j\mu (x), we get
j\mu (x) = O(x - \mu - 1/2).
This estimation allows us to conclude that there is a constant \kappa \mu related to \mu satisfying the
following inequality:
| j\mu (x)| \leq \kappa \mu x
- \mu - 1/2.
2. Main results. In this section, we state the Hardy’s and Miyachi’s theorems. We start by
Hardy’s theorem for the first Hankel – Clifford transform.
2.1. Hardy’s theorem for the first Hankel – Clifford transform.
Theorem 2.1. Let f be a measurable function on \BbbR such that
| f(x)| \leq C| x| \mu +1e - ax2
(2.1)
and
| (h1,\mu f)(y)| \leq C| y| \mu e -
y2
a . (2.2)
For some constants a, C > 0 the function f is a constant multiple of x\mu +1e - ax2
.
Proof. First, we have the function (h1,\mu f)(z) is well defined for all z. Moreover, by using the
estimated (4) of Lemma 1.3 and (2.1), for all z \in \BbbC , we have
\bigm| \bigm| (h1,\mu f)(z)\bigm| \bigm| \leq 2\mu | z| \mu
+\infty \int
0
\bigm| \bigm| \bigl( 2\surd xz
1
2
\bigr) - \mu
J\mu
\bigl(
2
\surd
xz
1
2
\bigr) \bigm| \bigm| | f(x)| dx \leq
\leq | z| \mu
+\infty \int
0
x\mu +1e - ax2
\infty \sum
k=0
\Bigl(
2| z|
1
2
\surd
x/2
\Bigr) 2k
\Gamma (k + \mu + 1)k!
dx \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
HARDY’S AND MIYACHI’S THEOREMS FOR THE FIRST HANKEL – CLIFFORD TRANSFORM 713
\leq | z| \mu
\infty \sum
k=0
\bigl(
| z|
1
2
\bigr) 2k
\Gamma (k + \mu + 1)k!
+\infty \int
0
x\mu +k+1e - ax2
dx \leq
\leq | z| \mu
\infty \sum
k=0
\bigl(
| z|
1
2
\bigr) 2k
\Gamma (k + \mu + 1)k!
\Gamma (\mu + k + 1)
ak+\mu +1
\leq
\leq C| z| \mu
\infty \sum
k=0
\Bigl(
| z|
1
2 /
\surd
a
\Bigr) 2k
k!
.
If | z| < 1, then
\bigm| \bigm| (h1,\mu f)(z)\bigm| \bigm| \leq C| z| \mu
\infty \sum
k=0
(1/
\surd
a)
2k
k!
= C| z| \mu e
1
a \leq
\leq C| z| \mu e
| z| \mu
a .
We put C \prime = Ce
1
a , so we have, if | z| < 1,\bigm| \bigm| (h1,\mu f)(z)\bigm| \bigm| \leq C \prime | z| \mu \leq C \prime | z| \mu e
| z| \mu
a .
If | z| \geq 1, then \bigm| \bigm| (h1,\mu f)(z)\bigm| \bigm| \leq C| z| \mu
\infty \sum
k=0
| 1/
\surd
a| 2k
k!
= C| z| \mu e
| z| 2
a .
So, for all z \in \BbbC , \bigm| \bigm| (h1,\mu f)(z)\bigm| \bigm| \leq C| z| \mu e
| z| 2
a .
Then \bigm| \bigm| z - \mu (h1,\mu f)(z)
\bigm| \bigm| \leq Ce
| z| 2
a .
From assumption (2.2) we obtain\bigm| \bigm| y - \mu (h1,\mu f)(y)
\bigm| \bigm| \leq Ce -
y2
a \forall y \in \BbbR .
Thus, z - \mu h1,\mu (f)(z) is an entire function, according to Lemma 1.1, \lambda - \mu h1,\mu (f)(\lambda ) must be a
multiple of e -
\lambda 2
a . Or we get
h1,\mu = h - 1
1,\mu ,
then f(\lambda ) is a multiple of \lambda \mu e -
\lambda 2
a .
Theorem 2.1 is proved.
2.2. Miyachi’s theorem for the first Hankel – Clifford transform.
Theorem 2.2. Let a > 0. We suppose that f is a function on \BbbR such that
x -
\mu
2
- 1
4 eax
2
f \in L1(\BbbR +) + L\infty (\BbbR +)
and
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
714 M. EL KASSIMI, S. FAHLAOUI
+\infty \int
- \infty
\mathrm{l}\mathrm{o}\mathrm{g}+
\Biggl(
| h1,\mu (f)(\lambda )\lambda - \mu
2
- 1
4 eb\lambda
2 |
A
\Biggr)
d\lambda < \infty
for some A, 0 < A < \infty . Then f is a constant multiple of x
\mu
2
+ 1
4 e - ax2
.
Proof. By the first assumption
x -
\mu
2
- 1
4 eax
2
f \in L1(\BbbR +) + L\infty (\BbbR +),
then there are two functions u \in L1(\BbbR +) and v \in L\infty (\BbbR +) such that
x -
\mu
2
- 1
4 eax
2
f(x) = u(x) + v(x)
and, thus,
f(x) = x
\mu
2
+ 1
4 e - ax2
u(x) + x
\mu
2
+ 1
4 e - ax2
v(x),
h1,\mu (f)(\lambda ) = h1,\mu (x
\mu
2
+ 1
4 e - ax2
u)(\lambda ) + h1,\mu
\bigl(
x
\mu
2
+ 1
4 e - ax2
v
\bigr)
(\lambda ).
If \lambda \in \BbbC , we have
\bigm| \bigm| h1,\mu \bigl( x\mu
2
+ 1
4 e - ax2
u
\bigr)
(\lambda )
\bigm| \bigm| \leq 2\mu | \lambda | \mu
+\infty \int
0
\bigm| \bigm| \bigl( 2\surd x\lambda
1
2
\bigr) - \mu
J\mu
\bigl(
2
\surd
x\lambda
1
2
\bigr) \bigm| \bigm| x\mu
2
+ 1
4 e - ax2 | u(x)| dx \leq
\leq C| \lambda | \mu
+\infty \int
0
\bigl(
2x
1
2\lambda
1
2
\bigr) - \mu - 1
2x
\mu
2
+ 1
4 e - ax2 | u(x)| dx \leq
\leq C| \lambda |
\mu
2
- 1
4
+\infty \int
0
x -
1
2
\mu - 1
4x
1
2
\mu + 1
4 e - ax2 | u(x)| dx \leq
\leq C| \lambda |
\mu
2
- 1
4
+\infty \int
0
e - ax2 | u(x)| dx \leq
\leq C| \lambda |
\mu
2
- 1
4 \leq C| \lambda |
\mu
2
- 1
4 e2Im(\lambda )2 ,
where C is a positive constant. Then\bigm| \bigm| h1,\mu \bigl( x\mu
2
+ 1
4 e - ax2
u
\bigr)
(\lambda )
\bigm| \bigm| \leq C| \lambda |
\mu
2
- 1
4 e2Im(\lambda )2 .
So, \bigm| \bigm| \lambda - \mu
2
+ 1
4h1,\mu
\bigl(
x
\mu
2
+ 1
4 e - ax2
u
\bigr)
(\lambda )
\bigm| \bigm| \leq Ce2Im(\lambda )2 ,
and we have
\bigm| \bigm| h1,\mu \bigl( x\mu
2
+ 1
4 e - ax2
v
\bigr)
(\lambda )
\bigm| \bigm| \leq 2\mu | \lambda | \mu
+\infty \int
0
\bigm| \bigm| \bigl( 2\surd x\lambda
1
2
\bigr) - \mu
J\mu
\bigl(
2
\surd
x\lambda
1
2
\bigr) \bigm| \bigm| x\mu
2
+ 1
4 e - ax2 | v(x)| dx \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
HARDY’S AND MIYACHI’S THEOREMS FOR THE FIRST HANKEL – CLIFFORD TRANSFORM 715
\leq | \lambda | \mu
+\infty \int
0
\bigl(
2x
1
2\lambda
1
2
\bigr) - \mu - 1
2x
\mu
2
+ 1
4 e - ax2 | v(x)| dx \leq
\leq | \lambda |
\mu
2
- 1
4
+\infty \int
0
x -
1
2
\mu - 1
4x
\mu
2
+ 1
4 e - ax2 | v(x)| dx \leq
\leq | \lambda |
\mu
2
- 1
4 \| v\| \infty
+\infty \int
0
e - ax2
uudx \leq
\leq C| \lambda |
\mu
2
- 1
4 \leq C| \lambda |
\mu
2
- 1
4 e2Im(\lambda )2 .
So, \bigm| \bigm| \bigm| \lambda - \mu
2
+ 1
4h1,\mu
\bigl(
x
\mu
2
+ 1
4 e - ax2
v
\bigr)
(\lambda )
\bigm| \bigm| \bigm| \leq Ce2Im(\lambda )2 ,
then \bigm| \bigm| \bigm| \lambda - \mu
2
+ 1
4h1,\mu (f)(\lambda )
\bigm| \bigm| \bigm| \leq Ce2Im(\lambda )2 .
Since \lambda - \mu
2
+ 1
4h1,\mu (f)(\lambda ) is an entire function, then, by the Lemma 1.2 and h1,\mu = h - 1
1,\mu , we can
obtain that f is a multiple of x\mu e - ax2
.
Theorem 2.2 is proved.
References
1. Abilov V. A., Abilova F. V. Approximation of functions by Fourier – Bessel sums // Izv. Vyssh. Uchebn. Zaved. Mat. –
2001. – № 8. – P. 3 – 9.
2. Betancor J. J. The Hankel – Clifford transformation on certain spaces of ultradistributions // Indian J. Pure and Appl.
Math. – 1989. – 20, № 6. – P. 583 – 603.
3. Cowling M., Price J. Generalizations of Heisenberg’s inequality // Harmonic Anal.: Lect. Notes Math. – Berlin:
Springer, 1983. – 992.
4. El Kassimi M. An Lp - Lq version of Morgan’s and Cowling – Price’s theorem for the first Hankel – Clifford
transform // Nonlinear Stud. (N.S.). – 2019. – 26, № 1.
5. Gray A., Matthews G. B., Macrobert T. M. A treatise on Bessel functions and their applications to physics. – London:
MacMillan, 1952.
6. Hardy G. H. A theorem concerning Fourier transforms // J. London Math. Soc. – 1933. – 8. – P. 227 – 231.
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8. Lebedev N. N. Special functions and their applications. – New York: Dover Publ. Inc., 1972.
9. M’endez P’erez J. M. R., Socas Robayna M. M. A pair of generalized Hankel – Clifford transformations and their
applications // J. Math. Anal. and Appl. – 1991. – 154, № 2. – P. 543 – 557.
10. Malgonde S. P., Bandewar S. R. On the generalized Hankel – Clifford transformation of arbitrary order // Proc. Indian
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11. Miyachi A. A generalization of theorem of Hardy // Harmonic Analysis. Semin. Izunagaoka. – Japon: Shizuoka-Ken,
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Received 04.07.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
|
| id | umjimathkievua-article-1469 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:17Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/73/e2521c32ecf5a7bf8895ec7d20a4cc73.pdf |
| spelling | umjimathkievua-article-14692019-12-05T08:56:08Z Hardy’s and Miyachi’s theorems for the first Hankel – Clifford transform Теореми Гардi та Мiячi про перше перетворення Ганкеля–Клiффорда El, Kassimi M. Fahlaoui, S. Ель, Кассимі М. Фахлауі, С. UDC 517.5 We present an analog of Hardy's and Miyachi's theorems for the first Hankel$\,--\,$,Clifford transform. УДК 517.5 Наведено аналог теорем Гарді та Міячі про перше перетворення Ганкеля$\,--\,$Кліффорда. Institute of Mathematics, NAS of Ukraine 2019-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1469 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 5 (2019); 710-715 Український математичний журнал; Том 71 № 5 (2019); 710-715 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1469/453 Copyright (c) 2019 El Kassimi M.; Fahlaoui S. |
| spellingShingle | El, Kassimi M. Fahlaoui, S. Ель, Кассимі М. Фахлауі, С. Hardy’s and Miyachi’s theorems for the first Hankel – Clifford transform |
| title | Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform |
| title_alt | Теореми Гардi та Мiячi про перше перетворення Ганкеля–Клiффорда |
| title_full | Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform |
| title_fullStr | Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform |
| title_full_unstemmed | Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform |
| title_short | Hardy’s and Miyachi’s theorems for the first
Hankel – Clifford transform |
| title_sort | hardy’s and miyachi’s theorems for the first
hankel – clifford transform |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1469 |
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