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

Journal of Inequalities and Applications20132013:29

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

Received: 17 November 2012

Accepted: 8 January 2013

Published: 22 January 2013

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.

Keywords

Bessel wavelet transformHankel transformHankel translationZemanian space

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 + ) , 1 p < , as the space of those real measurable functions ϕ on R + = ( 0 , ) for which
(1)
(2)

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

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 ) d μ ( z ) , 0 < x , y < ,
(3)
where
D ( x , y , z ) : = 0 j ( x t ) j ( y t ) j ( z t ) d μ ( 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 ) d μ ( 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 ) ¯ d μ ( 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 ( x u ) ϕ ( x ) d x , μ 1 / 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 ( b u ) f ˜ ( u ) ψ ˜ ( a u ) ¯ d u : = 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 , ρ R ,
(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 μ 1 / 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 ) ϵ u + 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 μ 1 / 2 , | ( b u ) μ n J μ + ν + n ( b u ) | 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 = a u
(23)
Therefore, (22) becomes
(24)
Suppose P is an integer not less than 2 μ + 2 n + 2 l + 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 ( ρ l ) / 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 μ R , 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 μ 1 / 2 .

Proof As in the proof of Theorem 2.1, using inequality (21), we have
(31)
Assume that 0 μ + 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 2 μ + 6 n + 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 = 2 ρ / d .

This completes the proof. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Applied Mathematics, Indian School of Mines, Dhanbad, India
(2)
Department of Mathematics, NERIST, Nirjuli, India

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Copyright

© Prasad et al.; licensee Springer 2013

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