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

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 29 (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.

## 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}_{\mu ,p}({\mathbb{R}}_{+}), 1\le p<\mathrm{\infty}, as the space of those real measurable functions *ϕ* on {\mathbb{R}}_{+}=(0,\mathrm{\infty}) for which

where d\mu (x)=1/{2}^{\mu}\mathrm{\Gamma}(\mu +1){x}^{2\mu +1}dx.

From [1, 4], the Hankel translation \varphi \in {L}_{\mu ,p}({\mathbb{R}}_{+}) is defined by

where

with j(x)={2}^{\mu}\mathrm{\Gamma}(\mu +1){x}^{-\mu}{J}_{\mu}(x).

From [5], in terms of the Hankel translation {\tau}_{y} and dilation {D}_{a} defined by {D}_{a}\varphi (x,y):=\varphi (x/a,y/a), we define the Bessel wavelet {\theta}_{b,a} by

Let \theta \in {L}_{\mu ,p}({\mathbb{R}}_{+}), then the Bessel wavelet transform of g\in {L}_{\mu ,q}({\mathbb{R}}_{+}), (1/p+1/q=1), is defined by

Now, using the Hankel transformation,

which is known to be an automorphism on the Zemanian space {H}_{\mu}({\mathbb{R}}_{+}), {\mathbb{R}}_{+}=(0,\mathrm{\infty}), consisting of all complex-valued infinitely differentiable functions *ϕ* on {\mathbb{R}}_{+} which satisfy

From (7), we have

Now, let us set

and

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

where \tilde{f}(u)=({h}_{\mu}f)(u).

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

## 2 The space {H}_{\mu}^{\omega}

Let *ω* be a continuous real-valued function defined on {\mathbb{R}}_{+}=(0,\mathrm{\infty}) such that \omega (0)=0 and

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}_{\mu}^{\omega}, where *f* and {h}_{\mu}f are smooth functions, and for every \mu \in {\mathbb{R}}_{+}, n\in {\mathbb{N}}_{0} and m is a positive real number,

On {H}_{\mu}^{\omega} we consider the topology generated by the family {\{{\alpha}_{m,n}^{\mu}\}}_{m\in {\mathbb{R}}_{+},n\in {\mathbb{N}}_{0}} of seminorms. From [1, 2], we have

In what follows, we study the Bessel wavelet transform {B}_{\psi} of infinite order on {H}_{\mu}^{\omega}. For this purpose, we define the symbol class {S}^{\rho ,\omega}.

**Definition 2.1** The function \tilde{\psi}(x):{C}^{\mathrm{\infty}}({\mathbb{R}}_{+})\to \mathbb{C} belongs to class {S}^{\rho ,\omega} if and only if \mathrm{\forall}m\in {\mathbb{R}}_{+}

where *c* is a constant and \tilde{\psi} 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}_{\mu}({\mathbb{R}}_{+}).

**Definition 2.2** The set of all infinitely differentiable functions ({B}_{\psi}\varphi )(b,a) on {\mathbb{R}}_{+}^{2} satisfying the condition

\mathrm{\forall}n,l\in {\mathbb{N}}_{0}, is denoted by {H}_{\mu}^{\omega}({\mathbb{R}}_{+}^{2}), where \mu ,m,{m}^{\prime}\in \mathbb{R}.

**Theorem 2.1** *The Bessel wavelet transform* {B}_{\psi}\varphi *is a continuous linear map of* {H}_{\mu}^{\omega}({\mathbb{R}}_{+}) *into* {H}_{\mu}^{\omega}({\mathbb{R}}_{+}^{2}) *for* \mu \ge -1/2.

*Proof* To complete the proof of the theorem, we need to show that ({B}_{\psi}\varphi )(b,a) satisfies (16). From the property (b) of the function \omega (u), it follows that to every \u03f5>0, there exists a constant c(\u03f5) such that

so that

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

Since for \mu \ge -1/2, |{(bu)}^{-\mu -n}{J}_{\mu +\nu +n}(bu)| is bounded on 0<b,u<\mathrm{\infty} by {Q}_{\mu}, the right-hand side of equation (21) can be estimated by

Therefore,

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

Therefore, (22) becomes

Suppose *P* is an integer not less than 2\mu +2n+2l+1, then

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

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

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

Therefore,

where {m}^{\prime}=2(\rho -l)/d. Now, choosing

we find that the last series is convergent. Therefore,

Hence, ({B}_{\psi}\varphi )\in {H}_{\mu}^{\omega}({\mathbb{R}}_{+}^{2}). □

## 3 The space {G}_{\mu}^{\omega}

The Bessel-differential operator {S}_{\mu} is defined by

From [[6], p.139] we know that for any \varphi \in {H}_{\mu}({\mathbb{R}}_{+}),

and

where {c}_{j} are constants depending only on *μ*.

Now, assume that *ω* is a function in *M*. *A* function \varphi \in {C}^{\mathrm{\infty}}({\mathbb{R}}_{+}) is said to be in the space {G}_{\mu}^{\omega} if for every \mu \in \mathbb{R}, n\in {\mathbb{N}}_{0}, and *m* is a positive real number,

The family {\{{A}_{m,n}^{\mu}\}}_{m\in \mathbb{R},n\in {\mathbb{N}}_{0}} of seminorms generates the topology of {G}_{\mu}^{\omega}.

**Definition 3.1** The set of all infinitely differentiable functions ({B}_{\psi}\varphi )(b,a) on {\mathbb{R}}_{+}^{2} satisfying the condition

is denoted by {G}_{\mu}^{\omega}({\mathbb{R}}_{+}^{2}), where \mu ,m,{m}^{\prime}\in \mathbb{R}.

**Theorem 3.1** *The Bessel wavelet transform* {B}_{\psi} *is a continuous linear map of* {G}_{\mu}^{\omega}({\mathbb{R}}_{+}) *into* {G}_{\mu}^{\omega}({\mathbb{R}}_{+}^{2}) *for* \mu \ge -1/2.

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

Assume that 0\le \mu +1/2<p, where *p* is a positive integer, then {b}^{\mu +1/2}\le {(1+b)}^{\mu +1/2}\le {(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\mu +6n+p+1, then the above expression can be estimated by

Using the inequality (1+u)\le {e}^{-l/d}{e}^{\omega (u)/d} from (14), the right-hand side of the above expression can be bounded by

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

Therefore,

as the infinite series can be made convergent by choosing

where {m}^{\prime}=2\rho /d.

This completes the proof. □

## References

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*Integral Transforms Spec. Funct.*2001, 11(1):61–72. 10.1080/10652460108819300Pathak RS, Prasad A:The pseudo-differential operator {h}_{\mu ,a} on some Gevrey spaces.

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*Trans. Am. Math. Soc.*1965, 116: 330–375.Pathak RS, Dixit MM: Bessel wavelet transform on certain function and distribution spaces.

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*Generalized Integral Transformations*. Interscience, New York; 1968.

## Acknowledgements

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

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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.

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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

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