Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets

We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet transform on $\mathbb{R}$. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.

Define the Dunkl-Sonine integral transform X α,β and its dual t X α,β , respectively, by Soltani has showed in [14] that the dual Dunkl-Sonine integral transform t X α,β is a transmutation operator between Λ α and Λ β on the Schwartz space S(R), i.e., it is an automorphism of S(R) satisfying the intertwining relation The same author [14] has obtained inversion formulas for the transform t X α,β involving pseudodifferential-difference operators and only valid on a restricted subspace of S(R). The purpose of this paper is to investigate the use of Dunkl wavelets (see [5]) to derive a new inversion of the dual Dunkl-Sonine transform on some Lebesgue spaces. For other applications of wavelet type transforms to inverse problems we refer the reader to [7,8] and the references therein.
The content of this article is as follows. In Section 2 we recall some basic harmonic analysis results related to the Dunkl operator. In Section 3 we list some basic properties of the Dunkl-Sonine integral trnsform and its dual. In Section 4 we give the definition of the Dunkl continuous wavelet transform and we establish for this transform a Calderón formula. By combining the results of the two previous sections, we obtain in Section 5 two new inversion formulas for the dual Dunkl-Sonine integral transform.
The Dunkl transform of order γ + 1/2 on R is defined for a function f in L 1 (R, |x| 2γ+1 dx) by It is known that the Dunkl transform F γ maps continuously and injectively L 1 (R, |x| 2γ+1 dx) into the space C 0 (R) (of continuous functions on R vanishing at infinity).
Two standard results about the Dunkl transform F γ are as follows.
An outstanding result about Dunkl kernels on R (see [12]) is the product formula where T x γ stand for the Dunkl translation operators defined by The Dunkl convolution of two functions f , g on R is defined by the relation Proposition 2.5 (see [13]). (2.5) (ii) For f ∈ L 1 (R, |x| 2γ+1 dx) and g ∈ L p (R, |x| 2γ+1 dx), p = 1 or 2, we have For more details about harmonic analysis related to the Dunkl operator on R the reader is referred, for example, to [9,10].
The next statement provides formulas relating harmonic analysis tools tied to Λ α with those tied to Λ β , and involving the operator t X α,β .
(i) Notice that by (3.4), (ii) It follows from (1.1) that for all λ ∈ C and x ∈ R.
Let ψ ∈ S(R). By using (3.3), (3.7) and Fubini's theorem, we have But an easy computation shows that t X α,β f − = t X α,β f − . Hence, This clearly yields the result. Definition 4.1. We say that a function g ∈ L 2 (R, |x| 2γ+1 dx) is a Dunkl wavelet of order γ, if it satisfies the admissibility condition Remark 4.2.
Definition 4.5. Let g ∈ L 2 (R, |x| 2γ+1 dx) be a Dunkl wavelet of order γ. We define for regular functions f on R, the Dunkl continuous wavelet transform by which can also be written in the form where * γ is the generalized convolution product given by (2.4), and g a (x) := g(−x/a), x ∈ R.
The Dunkl continuous wavelet transform has been investigated in depth in [5] in which precise definitions, examples, and a more complete discussion of its properties can be found. We look here for a Calderón formula for this transform. We start with some technical lemmas. Lemma 4.6. For all f, g ∈ L 2 (R, |x| 2γ+1 dx) and all ψ ∈ S(R) we have the identity where m γ is given by (2.2).
Proof . Suppose f 1 * γ f 2 ∈ L 2 (R, |x| 2γ+1 dx). By Lemma 4.6 and Theorem 2.3, we have for any ψ ∈ S(R), , then by Lemma 4.6 and Theorem 2.3, we have for any ψ ∈ S(R), which shows, in view of Theorem 2.4, that . This achieves the proof of Lemma 4.7.
A combination of Lemma 4.7 and Theorem 2.3 gives us the following.
where both sides are finite or infinite. Lemma 4.9. Let g ∈ L 2 (R, |x| 2γ+1 dx) be a Dunkl wavelet of order γ such that F γ g ∈ L ∞ (R). and

4)
and Proof . Using Schwarz inequality for the measure da a 4γ+5 we obtain By Theorem 2.3, Lemma 4.8, and Remark 4.4, we have The second assertion in (4.4) is easily checked. Let us calculate F γ (G ε,δ ). Fix x ∈ R. From Theorem 2.3 and Lemma 4.7 we get As |e γ (iz)| ≤ 1 for all z ∈ R (see [12]), we deduce by Theorem 2.3 that Hence, applying Fubini's theorem, we find that which completes the proof.
We can now state the main result of this section.
So (4.5) follows from the dominated convergence theorem.
Another pointwise inversion formula for the Dunkl wavelet transform, proved in [5], is as follows.
Theorem 4.11. Let g ∈ L 2 (R, |x| 2γ+1 dx) be a Dunkl wavelet of order γ. If both f and F γ f are in L 1 (R, |x| 2γ+1 dx) then we have where, for each x ∈ R, both the inner integral and the outer integral are absolutely convergent, but possibly not the double integral.

Inversion of the dual Dunkl-Sonine transform using Dunkl wavelets
From now on assume β > α > −1/2. In order to invert the dual Dunkl-Sonine transform, we need the following two technical lemmas.