Skip to main content

Continuity of the Bessel wavelet transform on certain Beurling-type function spaces

Abstract

In this paper, continuity of the Bessel wavelet transform of a suitable function ϕ in terms of an appropriate mother wavelet ψ is investigated on certain Beurling-type function spaces.

MSC:46F12, 26A33.

1 Introduction

Bessel wavelet transforms have applications in the study of boundary value problems on the half-line. The Hankel transform and pseudo-differential operators associated with the Bessel operator have been studied on some Beurling-type function spaces by [1, 2] and [3] respectively. The aim of the present paper is to study the continuity of the Bessel wavelet transform on certain Beurling-type function spaces.

Let us define L μ , p ( R + ), 1p<, as the space of those real measurable functions ϕ on R + =(0,) for which

(1)
(2)

where dμ(x)=1/ 2 μ Γ(μ+1) x 2 μ + 1 dx.

From [1, 4], the Hankel translation ϕ L μ , p ( R + ) is defined by

τ y ϕ(x):=ϕ(x,y):= 0 ϕ(z)D(x,y,z)dμ(z),0<x,y<,
(3)

where

D(x,y,z):= 0 j(xt)j(yt)j(zt)dμ(t)
(4)

with j(x)= 2 μ Γ(μ+1) x μ J μ (x).

From [5], in terms of the Hankel translation τ y and dilation D a defined by D a ϕ(x,y):=ϕ(x/a,y/a), we define the Bessel wavelet θ b , a by

θ b , a (x):= D a τ b θ(x)= D a θ(b,x)=θ(b/a,x/a):= 0 D(b/a,x/a,z)θ(z)dμ(z).
(5)

Let θ L μ , p ( R + ), then the Bessel wavelet transform of g L μ , q ( R + ), (1/p+1/q=1), is defined by

( B θ g)(b,a)= g ( t ) , θ b , a ( t ) = 0 g(t) θ b , a ( t ) ¯ dμ(t)
(6)
= 0 0 g ( t ) θ ( z ) ¯ D ( b / a , t / a , z ) d μ ( z ) d μ ( t ) = 0 ( b x / a ) 1 / 2 J μ ( b x / a ) x μ 1 / 2 b μ 1 / 2 a 2 μ + 1 × { 0 t μ + 1 / 2 g ( t ) ( t x / a ) 1 / 2 J μ ( t x / a ) d t × 0 z μ + 1 / 2 θ ( z ) ¯ ( z x ) 1 / 2 J μ ( z x ) d z } d x .
(7)

Now, using the Hankel transformation,

( h μ ϕ)(u)= 0 ( x u ) 1 / 2 J μ (xu)ϕ(x)dx,μ1/2
(8)

which is known to be an automorphism on the Zemanian space H μ ( R + ), R + =(0,), consisting of all complex-valued infinitely differentiable functions ϕ on R + which satisfy

γ m , k μ (ϕ)= sup x R + | x m ( x 1 D ) k x μ 1 / 2 ϕ(x)|<,m,k N 0 .
(9)

From (7), we have

Now, let us set

and

( B ψ f)(b,a)= b μ + 1 / 2 a μ 3 / 2 ( B θ g)(b,a)

to get the following convenient form of the Bessel wavelet transform:

( B ψ f)(b,a):= 0 ( b u ) 1 / 2 J μ (bu) f ˜ (u) ψ ˜ ( a u ) ¯ du:= h μ ( f ˜ ( u ) ψ ˜ ( a u ) ¯ ) (b),
(10)

where f ˜ (u)=( h μ f)(u).

From [[6], p.134], we shall use the following Leibnitz-type formula:

( x 1 D ) k ( x μ 1 / 2 ψ ϕ ) = s = 0 k ( k s ) ( x 1 D ) s ( x μ 1 / 2 ϕ ) ( x 1 D ) k s ψ.
(11)

2 The space H μ ω

Let ω be a continuous real-valued function defined on R + =(0,) such that ω(0)=0 and

(12)
(13)
(14)

for some real l and d>0.

The class of all such ω functions is denoted by M.

Now, assume that ω is a function in M. A function f is said to be in the space H μ ω , where f and h μ f are smooth functions, and for every μ R + , n N 0 and m is a positive real number,

α m , n μ (f)= sup u R + e m ω ( u ) | ( u 1 D ) n u μ 1 / 2 f(u)|<.
(15)

On H μ ω we consider the topology generated by the family { α m , n μ } m R + , n N 0 of seminorms. From [1, 2], we have

β m , n μ (f)= sup u R + e m ω ( u ) | ( u 1 D ) n u μ 1 / 2 ( h μ f)(u)|<,m R + ,n N 0 .
(16)

In what follows, we study the Bessel wavelet transform B ψ of infinite order on H μ ω . For this purpose, we define the symbol class S ρ , ω .

Definition 2.1 The function ψ ˜ (x): C ( R + )C belongs to class S ρ , ω if and only if m R +

e m ω ( x ) | ( x 1 / x ) p ψ ˜ ( x ) ¯ | c p , ρ ( 1 + x 2 ) ρ p ,p N 0 ,ρR,
(17)

where c is a constant and ψ ˜ denotes the Hankel transform of the basic wavelet ψ.

In this section, certain spaces of functions of Beurling-type are introduced on which Bessel wavelet transforms can be defined. First, we recall the definition of the Zemanian space H μ ( R + ).

Definition 2.2 The set of all infinitely differentiable functions ( B ψ ϕ)(b,a) on R + 2 satisfying the condition

γ n , l μ , m , m ( B ψ ϕ)= sup a , b e m ω ( b ) m ω ( a ) | ( b 1 D ) n ( a 1 D ) l b μ 1 / 2 ( B ψ ϕ)(b,a)|<,
(18)

n,l N 0 , is denoted by H μ ω ( R + 2 ), where μ,m, m R.

Theorem 2.1 The Bessel wavelet transform B ψ ϕ is a continuous linear map of H μ ω ( R + ) into H μ ω ( R + 2 ) for μ1/2.

Proof To complete the proof of the theorem, we need to show that ( B ψ ϕ)(b,a) satisfies (16). From the property (b) of the function ω(u), it follows that to every ϵ>0, there exists a constant c(ϵ) such that

ω(u)ϵu+c(ϵ);
(19)

so that

e m ω ( u ) e m c ( ϵ ) ν = 0 ( m ϵ ) ν ν ! u ν .
(20)

Using equation (10) and the technique of Zemanian [[6], p.141], we can write

(21)

Since for μ1/2, | ( b u ) μ n J μ + ν + n (bu)| is bounded on 0<b,u< by Q μ , the right-hand side of equation (21) can be estimated by

Therefore,

(22)

In view of estimate (17), we have for t=au

(23)

Therefore, (22) becomes

(24)

Suppose P is an integer not less than 2μ+2n+2l+1, then

u 2 μ + 2 n + ν + 1 + 2 l ( 1 + u ) P + ν .
(25)

Using (25), the right-hand side of equation (24) can be bounded by

Using inequality (14), the above expression is bounded by

(26)

Using property (16), we can estimate the right-hand side of (26) by

Therefore,

where m =2(ρl)/d. Now, choosing

ϵ< { max 0 r ν β ( P + 2 ρ + ν ) / d , ( ν r ) μ ( ϕ ) } 1 / ν ( ν ! m e l / d ( 1 + c ) ) 1 ,

we find that the last series is convergent. Therefore,

e m ω ( b ) m ω ( a ) | ( b 1 D ) n ( a 1 D ) l b μ 1 / 2 ( B ψ ϕ)(b,a)|<.

Hence, ( B ψ ϕ) H μ ω ( R + 2 ). □

3 The space G μ ω

The Bessel-differential operator S μ is defined by

S μ , x = d 2 d x 2 + ( 1 4 μ 2 ) 4 x 2 .
(27)

From [[6], p.139] we know that for any ϕ H μ ( R + ),

h μ ( S μ ϕ)= y 2 h μ ϕ
(28)

and

S μ , x r ϕ(x)= j = 0 r c j x 2 j + μ + 1 / 2 ( x 1 D ) r + j ( x μ 1 / 2 ϕ ( x ) ) ,
(29)

where c j are constants depending only on μ.

Now, assume that ω is a function in M. A function ϕ C ( R + ) is said to be in the space G μ ω if for every μR, n N 0 , and m is a positive real number,

A m , n μ (ϕ)= sup x ( R + ) e m ω ( x ) | S μ n ϕ(x)|<.
(30)

The family { A m , n μ } m R , n N 0 of seminorms generates the topology of G μ ω .

Definition 3.1 The set of all infinitely differentiable functions ( B ψ ϕ)(b,a) on R + 2 satisfying the condition

δ n μ , m , m ( B ψ ϕ)= sup a , b e m ω ( b ) m ω ( a ) | S μ , b n ( B ψ ϕ)(b,a)|<,n N 0 ,

is denoted by G μ ω ( R + 2 ), where μ,m, m R.

Theorem 3.1 The Bessel wavelet transform B ψ is a continuous linear map of G μ ω ( R + ) into G μ ω ( R + 2 ) for μ1/2.

Proof As in the proof of Theorem 2.1, using inequality (21), we have

(31)

Assume that 0μ+1/2<p, where p is a positive integer, then b μ + 1 / 2 ( 1 + b ) μ + 1 / 2 ( 1 + b ) p , and the right-hand side of equation (31) is bounded by

Using Zemanian’s technique, equation (10) and the Leibnitz-type formula (11), the last expression can be estimated by

In view of estimate (17), the above expression can be bounded by

Suppose that N is a positive integer not less than 2μ+6n+p+1, then the above expression can be estimated by

Using the inequality (1+u) e l / d e ω ( u ) / d from (14), the right-hand side of the above expression can be bounded by

(32)

Using property (16), we can estimate the right-hand side of (32) by

Therefore,

as the infinite series can be made convergent by choosing

ϵ< { ( 1 + c ) k max 0 s k β ( N + ν + 2 ρ ) / d , ( k s ) μ ( ϕ ) } 1 / ν ( ν ! m e l / d ) 1 ,

where m =2ρ/d.

This completes the proof. □

References

  1. Belhadj M, Betancor JJ: Hankel transformation and Hankel convolution of tempered distributions. Rocky Mt. J. Math. 2001, 3(4):1171–1203.

    Article  MathSciNet  Google Scholar 

  2. Pathak RS, Shrestha KK: The Hankel transform of Gevrey ultradistributions. Integral Transforms Spec. Funct. 2001, 11(1):61–72. 10.1080/10652460108819300

    Article  MATH  MathSciNet  Google Scholar 

  3. Pathak RS, Prasad A:The pseudo-differential operator h μ , a on some Gevrey spaces. Indian J. Pure Appl. Math. 2003, 34(8):1225–1236.

    MATH  MathSciNet  Google Scholar 

  4. Haimo DT: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 1965, 116: 330–375.

    Article  MATH  MathSciNet  Google Scholar 

  5. Pathak RS, Dixit MM: Bessel wavelet transform on certain function and distribution spaces. J. Anal. Appl. 2003, 1: 65–83.

    MATH  MathSciNet  Google Scholar 

  6. Zemanian AH: Generalized Integral Transformations. Interscience, New York; 1968.

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by Indian School of Mines, Dhanbad, under grant No. 613002 /ISM JRF/Acad/2009.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ashutosh Mahato.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AP has identified the problems and suggested the solution, AM participated in the proof of the Theorems and MMD participated in the solution to find the Bessel wavelet transform. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Prasad, A., Mahato, A. & Dixit, M. Continuity of the Bessel wavelet transform on certain Beurling-type function spaces. J Inequal Appl 2013, 29 (2013). https://doi.org/10.1186/1029-242X-2013-29

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-29

Keywords

  • Bessel wavelet transform
  • Hankel transform
  • Hankel translation
  • Zemanian space