Generalized Bessel transform of (β, γ)-generalized Bessel Lipschitz functions

Generalized Bessel transform of ð; Þ-generalized Bessel Lipschitz functions

Dedicated to Professor Franc¸ois Rouvie`re for 70th birthday

By Radouan DAHERand Mohamed ELHAMMA

Departement of Mathematics, Faculty of Sciences A€n Chock, University of Hassan II, BP 5366 Maarif, Casablanca, Morocco

(Communicated by Kenji FUKAYA,M.J.A., May 12, 2015)

Abstract: In this paper, we prove an analog of Younis’s theorem 5.2 in [4] for the generalized Fourier-Bessel transform on the Half line for functions satisfying the ð; Þ-gener- alized Bessel Lipschitz condition in the spaceL²_;n.

Key words: Generalized Fourier-Bessel transform; generalized translation operator.

1. Introduction and preliminaries. Younis ([4], Theorem 5.2) characterized the set of functions inL²ðRÞsatisfying the Dini-Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely we have

Theorem 1.1 ([4], Theorem 5.2). Let f 2 L²ðRÞ. Then the following are equivalents:

1. kfð:þhÞ fð:Þk_L²_ðRÞ¼Oð ^h

ðlog_h¹ÞÞas h!0;0<

<1; >0, 2. R

jjrjFðfÞðÞj²d¼Oðr²ðlogrÞ²Þ as r! þ1,

whereF stands for the Fourier transform of f.

The main aim of this paper is to establish an analog of Theorem 1.1 in the generalized Fourier- Bessel transform. We point out that similar results have been established in the Dunkl transform [3].

We briefly overview the theory of generalized Fourier-Bessel transform and related harmonic analysis (see [2]).

Consider the second-order singular differential operator on the half line

BfðxÞ ¼ d²fðxÞ

dx² þð2þ1Þ x

dfðxÞ

dx 4nðþnÞ x² fðxÞ;

where >¹₂ and n¼0;1;2;. . .. For n¼0, we obtain the classical Bessel operator

BfðxÞ ¼d²fðxÞ

dx² þð2þ1Þ x

dfðxÞ dx :

For >¹₂ and n¼0;1;2;. . ., let M be the map defined by

MfðxÞ ¼x²ⁿfðxÞ:

LetL²_;n be the class of measurable functionsf on½0;1½for which

kfk_2;;n ¼ kM¹fk_2;þ2n<1;

where

kfk_2; ¼ Z þ1

0

jfðxÞj²x^2þ1dx

¹⁼²

:

For >¹₂, we introduce the normalized spherical Bessel function jdefined by

jðzÞ ¼ðþ1ÞX¹

k¼0

ð1Þ^k k!ðkþpþ1Þ

z 2

2k

; z2C;

where ðxÞ is the gamma-function. The function y¼jðxÞsatisfies the differential equation

Byþy¼0

with the initial conditions yð0Þ ¼1 and y⁰ð0Þ ¼0.

The functionjðxÞis infinitely differentiable, even, and, moreover entire analytic.

Lemma 1.2. For x2R^þ the following in- equalities are fulfilled.

1. jjðxÞj 1, 2. j1jðxÞj x,

3. j1jðxÞj c with x1, where c >0 is a certain constant which depends only on.

Proof (See [1]).

For2Cand x2R, put

’ðxÞ ¼x²ⁿjþ2nðxÞ:

From [2] recall the following properties.

doi: 10.3792/pjaa.91.85

#2015 The Japan Academy

2000 Mathematics Subject Classification. Primary 42B10.

No. 6] Proc. Japan Acad.,91, Ser. A (2015) 85

Proposition 1.3. 1. ’satisfies the differ- ential equation

B’¼ ²’:

2. For all2C, andx2R j’ðxÞj x²ⁿe^jImjjxj:

The generalized Fourier-Bessel transform we call the integral from [2]

FBðfÞðÞ ¼ Z þ1

0

fðxÞ’ðxÞx^2þ1dx; 0; f2L¹_;n:

Let f2L¹_;n, the inverse generalized Fourier- Bessel transform is given by the formula

fðxÞ ¼ Z þ1

0

FBðfÞðÞ’ðxÞdþ2nðÞ;

where

d_þ2nðÞ ¼ 1

4^þ2nððþ2nþ1ÞÞ^22þ4nþ1d:

From [2], we have

Theorem 1.4. 1. For every f 2L¹_;n\ L²_;nwe have the Plancherel formula

Z þ1 0

jfðxÞj²x^2þ1dx

¼ Z þ1

0

jFBðfÞðÞj²d_þ2nðÞ:

2. The generalized Fourier-Bessel transform FB extends uniquely to an isometric isomor- phism fromL²_;nonto L²ð½0;þ1½; þ2nÞ.

Define the generalized translation operatorT_h, h >0by the relation

ThfðxÞ ¼ ðxhÞ²ⁿ_þ2n^h ðM¹fÞðxÞ; x0;

where_þ2n^h are the Bessel translation operators of orderþ2ndefined by

^hfðxÞ ¼c Z

0

fð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²þh²2xhcost

p Þsin²tdt;

where c¼

Z 0

sin²tdt

1

¼ ðþ1Þ ð¹₂Þðþ¹₂Þ: By Proposition 3.2 in [2], we have

FBðThfÞðÞ ¼’ðhÞFBðfÞðÞ; f2L²_;n: ð1Þ

2. Main result. In this section we give the main result of this paper. We need first to define the

ð; Þ-generalized Bessel Lipschitz class

Definition 2.1. Let 2 ð0;1Þ and 0. A functionf2L²_;nis said to be in theð; Þ-general- ized Bessel Lipschitz class, denoted by BLipð;2; Þ, if

kThfð:Þ þT_hfð:Þ 2h²ⁿfð:Þk_2;;n

¼O h^þ2n ðlog¹_hÞ

!

ash!0:

Theorem 2.2. Letf2L²_;n. Then the follow- ing are equivalents

1. f2BLipð;2; Þ, 2. Rþ1

r jFBðfÞðÞj²dþ2nðÞ ¼O

r² ðlogrÞ²

asr! þ1.

Proof. 1)¼)2) Assume thatf 2BLipð;2; Þ.

Then

kThfð:Þ þT_hfð:Þ 2h²ⁿfð:Þk_2;;n

¼O h^þ2n ðlog¹_hÞ

!

ash!0:

Formula (1), we have

FBðThfþT_hf2h²ⁿfÞðÞ

¼’ðhÞ þ’ðhÞ 2h²ⁿ

FBðfÞðÞ

¼ ðh²ⁿjþ2nðhÞ þ ðhÞ²ⁿjþ2nðhÞ 2h²ⁿÞFBðfÞðÞ

¼2h²ⁿðjþ2nðhÞ 1ÞFBðfÞðÞ:

Plancherel identity gives kThfð:Þ þThfð:Þ 2h²ⁿfð:Þk²_2;;n

¼ Z þ1

0

4h⁴ⁿj1j_þ2nðhÞj²jFBðfÞðÞj²d_þ2nðÞ:

If2 ½_h¹;²_h, thenh1and (3) of Lemma 1.2 implies that

1 1

c²j1j_þ2nðhÞj²: Then

Z ²_h

1 h

jFBðfÞðÞj²dþ2nðÞ

1 c²

Z _h²

1 h

j1j_þ2nðhÞj²jFBðfÞðÞj²d_þ2nðÞ

1 c²

Z þ1 0

j1j_þ2nðhÞj²jFBðfÞðÞj²d_þ2nðÞ

86 R. DAHERand M. ELHAMMA [Vol. 91(A),

1 c²

1

4h⁴ⁿkThfð:Þ þT_hfð:Þ 2h²ⁿfð:Þk²_2;;n

¼O h² ðlog¹_hÞ²

! :

We obtain Z 2r

r

jFBðfÞðÞj²d_þ2nðÞ

¼O r² ðlogrÞ²

!

asr! þ1:

Thus these exists C >0such that Z 2r

r

jFBðfÞðÞj²d_þ2nðÞ C r² ðlogrÞ²: So that

Z þ1 r

jFBðfÞðÞj²d_þ2nðÞ

¼ Z 2r

r

þ Z 4r

2r

þ Z 8r

4r

þ. . .

jFBðfÞðÞj²d_þ2nðÞ

C r^2k

ðlogrÞ²þC ð2rÞ^2k

ðlog 2rÞ² þC ð4rÞ^2k ðlog 4rÞ²þ. . .

C r^2k

ðlogrÞ²ð1þ2^2kþ ð2^2kÞ²þ ð2^2kÞ³þ. . .Þ

CK r^2k ðlogrÞ²;

whereK¼ ð12^2kÞ¹ since 2^2k <1.

This prove that Z þ1

r

jFBðfÞðÞj²dþ2nðÞ

¼O r^2k ðlogrÞ²

!

asr! þ1:

2) ¼)1) Suppose now that Z þ1

r

jFBðfÞðÞj²d_þ2nðÞ

¼O r^2k ðlogrÞ²

!

asr! þ1:

We write Z þ1

0

j1j_þ2nðhÞj²jFBðfÞðÞj²d_þ2nðÞ ¼I1þI2;

where

I1 ¼ Z _h¹

0

j1jþ2nðhÞj²jFBðfÞðÞj²dþ2nðÞ

and I2 ¼

Z þ1 1 h

j1jþ2nðhÞj²jFBðfÞðÞj²dþ2nðÞ:

Estimate the summandsI1 and I2.

From inequality (1) of Lemma 1.2, we have I2 ¼

Z þ1 1 h

j1jþ2nðhÞj²jFBðfÞðÞj²dþ2nðÞ

4 Z þ1

1 h

jFBðfÞðÞj²d_þ2nðÞ

¼O h² ðlog¹_hÞ²

! : Then

4h⁴ⁿI2 ¼O h^2þ4n ðlog¹_hÞ²

! :

To estimate I1, we use the inequality (2) of Lemma 1.2. Set

ðxÞ ¼ Z þ1

x

jFBðfÞðÞj²dþ2nðÞ:

Using integration by parts, we obtain I1 C1h²

Z ¹_h

0

s^{2 0}ðsÞds

C1

1

h þ2C1h² Z ¹_h

0

s ðsÞds C2h²

Z 1=h 0

ss²ðlogsÞ²ds

C₃h² log1 h

2

;

whereC1,C2 andC3 are positive constants.

Then

4h⁴ⁿI1 ¼O h^2þ4n ðlog¹_hÞ²

! :

Furthermore, we have

kThfð:Þ þT_hfð:Þ 2h²ⁿfð:Þk_2;;n

¼O h^2þ4n ðlog¹_hÞ²

!

as h!0

and this ends the proof.

No. 6] Generalized Bessel transform ofð; Þ-generalized Bessel Lipschitz functions 87

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Uchebn. Zaved. Mat. 2001, no. 8, 3–9; trans- lation in Russian Math. (Iz. VUZ) 45 (2001), no. 8, 1–7 (2002).

[ 2 ] R. F. Al Subaie and M. A. Mourou, The contin- uous wavelet transform for A Bessel type

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88 R. DAHERand M. ELHAMMA [Vol. 91(A),