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

(1)

(2)

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

$${\tau }_y\varphi (x):=\varphi (x,y):={\int }_0^{\mathrm{\infty }}\varphi (z)D(x,y,z)d\mu (z),0<x,y<\mathrm{\infty },$$

(3)

where

$$D(x,y,z):={\int }_0^{\mathrm{\infty }}j(xt)j(yt)j(zt)d\mu (t)$$

(4)

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

$${\theta }_{b,a}(x):=D_a{\tau }_b\theta (x)=D_a\theta (b,x)=\theta (b/a,x/a):={\int }_0^{\mathrm{\infty }}D(b/a,x/a,z)\theta (z)d\mu (z).$$

(5)

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

$$(B_{\theta }g)(b,a)=〈g(t),{\theta }_{b,a}(t)〉={\int }_0^{\mathrm{\infty }}g(t)\overline{{\theta }_{b,a}(t)}d\mu (t)$$

(6)

$$\begin{array}{rc}=& {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}g(t)\overline{\theta (z)}D(b/a,t/a,z)d\mu (z)d\mu (t)\\ =& {\int }_0^{\mathrm{\infty }}{(bx/a)}^{1/2}{J}_{\mu }(bx/a)x^{-\mu -1/2}{b}^{-\mu -1/2}{a}^{2\mu +1}\\ \times \{{\int }_0^{\mathrm{\infty }}{t}^{\mu +1/2}g(t){(tx/a)}^{1/2}{J}_{\mu }(tx/a)dt\\ \times {\int }_0^{\mathrm{\infty }}{z}^{\mu +1/2}\overline{\theta (z)}{(zx)}^{1/2}{J}_{\mu }(zx)dz\}dx.\end{array}$$

(7)

Now, using the Hankel transformation,

$$(h_{\mu }\varphi )(u)={\int }_0^{\mathrm{\infty }}{(xu)}^{1/2}{J}_{\mu }(xu)\varphi (x)dx,\mu \ge -1/2$$

(8)

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

$${\gamma }_{m,k}^{\mu }(\varphi )=\underset{x\in {\mathbb{R}}_{+}}{sup}|x^m{\left(x^{-1}D\right)}^k{x}^{-\mu -1/2}\varphi (x)|<\mathrm{\infty },\mathrm{\forall }m,k\in {\mathbb{N}}_0.$$

(9)

From (7), we have

Now, let us set

and

$$(B_{\psi }f)(b,a)=b^{\mu +1/2}{a}^{-\mu -3/2}(B_{\theta }g)(b,a)$$

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

$$(B_{\psi }f)(b,a):={\int }_0^{\mathrm{\infty }}{(bu)}^{1/2}{J}_{\mu }(bu)\tilde{f}(u)\overline{\tilde{\psi }(au)}du:=h_{\mu }(\tilde{f}(u)\overline{\tilde{\psi }(au)})(b),$$

(10)

where $\tilde{f}(u)=(h_{\mu }f)(u)$.

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

$${\left(x^{-1}D\right)}^k\left(x^{-\mu -1/2}\psi \varphi \right)=\sum _{s=0}^k\left(\genfrac{}{}{0ex}{}{k}{s}\right){\left(x^{-1}D\right)}^s\left(x^{-\mu -1/2}\varphi \right){\left(x^{-1}D\right)}^{k-s}\psi .$$

(11)

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

(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 $H_{\mu }^{\omega }$, 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,

$${\alpha }_{m,n}^{\mu }(f)=\underset{u\in {\mathbb{R}}_{+}}{sup}{e}^{m\omega (u)}|{\left(u^{-1}D\right)}^n{u}^{-\mu -1/2}f(u)|<\mathrm{\infty }.$$

(15)

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

$${\beta }_{m,n}^{\mu }(f)=\underset{u\in {\mathbb{R}}_{+}}{sup}{e}^{m\omega (u)}|{\left(u^{-1}D\right)}^n{u}^{-\mu -1/2}(h_{\mu }f)(u)|<\mathrm{\infty },\mathrm{\forall }m\in {\mathbb{R}}_{+},n\in {\mathbb{N}}_0.$$

(16)

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 $\mathrm{\forall }m\in {\mathbb{R}}_{+}$

$$e^{m\omega (x)}\left|{(x^{-1}\partial /\partial x)}^p\overline{\tilde{\psi }(x)}\right|\le c^{p,\rho }{(1+x^2)}^{\rho -p},\mathrm{\forall }p\in {\mathbb{N}}_0,\rho \in \mathbb{R},$$

(17)

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

$${\gamma }_{n,l}^{\mu ,m,m^{\prime }}(B_{\psi }\varphi )=\underset{a,b}{sup}{e}^{m\omega (b)-m^{\prime }\omega (a)}|{\left(b^{-1}D\right)}^n{\left(a^{-1}D\right)}^l{b}^{-\mu -1/2}(B_{\psi }\varphi )(b,a)|<\mathrm{\infty },$$

(18)

$\mathrm{\forall }n,l\in {\mathbb{N}}_0$, 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 $ϵ>0$, there exists a constant $c(ϵ)$ such that

$$\omega (u)\le ϵu+c(ϵ);$$

(19)

so that

$$e^{m\omega (u)}\le e^{mc(ϵ)}\sum _{\nu =0}^{\mathrm{\infty }}\frac{{(mϵ)}^{\nu }}{\nu !}{u}^{\nu }.$$

(20)

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

(21)

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,

(22)

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

(23)

Therefore, (22) becomes

(24)

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

$$u^{2\mu +2n+\nu +1+2l}\le {(1+u)}^{P+\nu }.$$

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

$$ϵ<{\{\underset{0\le r\le \nu }{max}{\beta }_{(P+2\rho +\nu )/d,(\nu -r)}^{\mu }(\varphi )\}}^{-1/\nu }{(\nu !m{e}^{-l/d}(1+c))}^{-1},$$

we find that the last series is convergent. Therefore,

$$e^{m\omega (b)-m^{\prime }\omega (a)}|{\left(b^{-1}D\right)}^n{\left(a^{-1}D\right)}^l{b}^{-\mu -1/2}(B_{\psi }\varphi )(b,a)|<\mathrm{\infty }.$$

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

$$S_{\mu ,x}=\frac{d^2}{d{x}^2}+\frac{(1-4{\mu }^2)}{4x^2}.$$

(27)

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

$$h_{\mu }(S_{\mu }\varphi )=-y^2{h}_{\mu }\varphi$$

(28)

and

$$S_{\mu ,x}^r\varphi (x)=\sum _{j=0}^r{c}_j{x}^{2j+\mu +1/2}{\left(x^{-1}D\right)}^{r+j}(x^{-\mu -1/2}\varphi (x)),$$

(29)

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,

$$A_{m,n}^{\mu }(\varphi )=\underset{x\in ({\mathbb{R}}_{+})}{sup}{e}^{m\omega (x)}|S_{\mu }^n\varphi (x)|<\mathrm{\infty }.$$

(30)

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

$${\delta }_n^{\mu ,m,m^{\prime }}(B_{\psi }\varphi )=\underset{a,b}{sup}{e}^{m\omega (b)-m^{\prime }\omega (a)}|S_{\mu ,b}^n(B_{\psi }\varphi )(b,a)|<\mathrm{\infty },\mathrm{\forall }n\in {\mathbb{N}}_0,$$

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

(31)

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

(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

$$ϵ<{\{{(1+c)}^k\underset{0\le s\le k}{max}{\beta }_{(N+\nu +2\rho )/d,(k-s)}^{\mu }(\varphi )\}}^{-1/\nu }{(\nu !m{e}^{-l/d})}^{-1},$$

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

This completes the proof. □