The Continuous Wavelet Transform for A Bessel Type Operator on the Half Line

We consider a singular differential operator ∆ on the half line which generalizes the Bessel operator. Using harmonic analysis tools corresponding to ∆, we construct and investigate a new continuous wavelet transform on [0,∞[ tied to ∆. We apply this wavelet transform to invert an intertwining operator between ∆ and the second derivative operator d/dx.


Introduction
Consider the second-order singular differential operator on the half line where α > −1/2 and n = 0, 1, ... . For n = 0, we regain the differential operator which is referred to as the Bessel operator of order α.
A well known harmonic analysis on the half line generated by the Bessel operator L α , is amply and brilliantly exposed by Trimeche in [14]. Selected excerpts of this harmonic analysis are presented in Section 2.
The authors have showed in [1] that the integral transform X (f )(x) = 2 Γ(α + 2n + 1) √ π Γ(α + 2n + 1/2) is a topological isomorphism between two suitable functional spaces, satisfying the intertwining relation Through the intertwining operator X , a completely new commutative harmonic analysis on the half line related to the differential operator ∆, was initiated. A summary of this harmonic analysis is provided in Section 3. The main contribution of this work is to extend the classical theory of wavelets to the differential operator ∆. More explicitly, we call generalized wavelet each function g in a suitable functional space, satisfying the admissibility condition where F ∆ denotes the generalized Fourier transform related to ∆ given by with φ λ (x) = x 2n j α+2n (λx), j α+2n being the normalized spherical Bessel function of index α + 2n. Starting from a single generalized wavelet g we construct by dilation and translation a family of generalized wavelets by putting where g a (x) = g(x/a) and T b stand for the generalized translation operators tied to the differential operator ∆.
Thereby, the generalized continuous wavelet transform associated with ∆ is defined for regular functions In Section 4, we exhibit a relationship between the generalized and Bessel continuous wavelet transforms. Such a relationship enables us to establish for the generalized continuous wavelet transform a Plancherel formula, a pointwise reconstruction formula and a Calderon reproducing formula.
In Section 5, we exploit the intertwining operator X to express the generalized continuous wavelet transform in terms of the classical one. As a consequence, we derive new inversion formulas for dual operator t X of X . For examples of use of wavelet type transforms in inverse problems the reader is referred to [6,10,11,12,13] and the references therein.
In the classical framework, the notion of wavelets was first introduced by J. Morlet a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by A. Grossmann and J. Morlet in [5]. The harmonic analyst Y. Meyer and many other mathematicians became aware of this theory and they recognized many classical results inside it (see [2,8,9]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [3,4,7] and the references therein).

Preliminaries
In the present section we recapitulate some facts about harmonic analysis related to the Bessel operator L α . We cite here, as briefly as possible, only those properties actually required for the discussion. For more details we refer to [14].

Note 2.1 Throughout this section assume
The Fourier-Bessel transform of order α is defined for a function f ∈ L 1 α by where j α is the normalized spherical Bessel function of index α defined by Proposition 2.1 (i) The Fourier-Bessel transform F α maps continuously and injectively L 1 α into the space (iv) The Fourier-Bessel transform F α extends uniquely to an isometric isomorphism from L 2 α onto L 2 ([0, ∞[, µ α ). The inverse transform is given by where the integral converges in L 2 α .
The Bessel translation operators τ x α , x ≥ 0, are defined by where For x, y > 0, a change of variables yields with The Bessel convolution product of two functions f, g on [0, ∞[ is defined by the relation

Definition 2.1 We say that a function g ∈ L 2
α is a Bessel wavelet of order α, if it satisfies the admissibility condition where a > 0, b ≥ 0, and 198 The Continuous Wavelet Transform for A Bessel Type Operator on the Half Line The Bessel continuous wavelet transform has been investigated in depth in [14] from which we recall the following basic properties.
belongs to L 2 α and satisfies for almost all x ≥ 0.

Generalized Fourier transform
For λ ∈ C and x ∈ R, put where j α+2n is the normalized Bessel function of index α + 2n given by (2). From [1] recall the following properties.

Proof. By
for almost all x ≥ 0.
(ii) The generalized Fourier transform F ∆ extends uniquely to an isometric isomorphism from L 2 α,n onto The inverse transform is given by where the integral converges in L 2 α,n .
which gives (i). The proof of (ii) is standard.

Generalized convolution product
Definition 3.2 Define the generalized translation operators T x , x ≥ 0, by the relation where τ x α+2n are the Bessel translation operators of order α + 2n given by (4).
Then for all x ≥ 0, the function T x f belongs to L p α,n , and (ii) For f ∈ L p α,n , p = 1 or 2, we have α,n and g ∈ L q α,n then f #g ∈ L r α,n and ∥f #g∥ r,α,n ≤ ∥f ∥ p,α,n ∥g∥ q,α,n .

Transmutation operators
Note 3. 2 We denote by E(R) the space of C ∞ even functions on R, provided with the topology of compact convergence for all derivatives. For a > 0, D a (R) designates the space of C ∞ even functions on R, which are supported in [−a, a], equipped with the topology induced by E(R). Put D(R) = ∪ a>0 D a (R) endowed with the inductive limit topology. Let E n (R) (resp. D n (R)) stand for the subspace of E(R) (resp. D(R)) consisting of functions f such that f (0) = · · · = f (2n−1) (0) = 0.

Definition 3.4 For a locally bounded function f on
where a α+2n is given by (5).

Remark 3.5 (i) For n = 0, X reduces to the Riemann-Liouville integral transform of order α given by
(ii) It is easily checked that (iii) Due to (14) and (20) we have

Remark 3.6 (i) For n = 0, t X is just the Weyl integral transform of order α given by
(ii) It is easily seen that where F c is the cosine transform given by (v) Let f, g ∈ L 1 α,n . Then where * is the symmetric convolution product on [0, ∞[ defined by (iv) By (16), (23) and [14,Equation (5.II.14)] we have (vi) By (19), (21), (23) and [14,Equation (7.IV.9)] we have This achieves the proof. X and t X are intertwining operators between ∆ and the second derivative operator d 2 /dx 2 by virtue of the following theorem proved in [1].

Theorem 3.3 (i) The integral transform X is an isomorphism from E(R) onto E n (R) satisfying the intertwining relation
(ii) The integral transform t X is an isomorphism from D n (R) onto D(R) satisfying the intertwining relation
(ii) By (9), (16) and (26), g ∈ L 2 α,n is a generalized wavelet if and only if, M −1 g is a Bessel wavelet of order α + 2n , and we have where g a is given by (12) and T b are the generalized translation operators defined by (17).

Definition 4.2
Let g ∈ L 2 α,n be a generalized wavelet. We define for regular functions f on [0, ∞[, the generalized continuous wavelet transform by which can also be written in the form where # is the generalized convolution product given by (18).
which ends the proof.
Proof. By (27), (29) and (32) we have The result is then a direct consequence of and Theorem 2.1(ii) .

Theorem 4.3 (inversion formula)
then we have for almost all x ≥ 0.

Inversion of the intertwining operator t X through the generalized wavelet transform
To obtain inversion formulas for t X involving generalized wavelets, we have to establish some preliminary lemmas.
as λ → 0. Then X g ∈ L 2 α,n is a generalized wavelet and Proof. By (35) and Lemma 5.1 we see that X g ∈ L 2 α,n , F ∆ (X g) is bounded and Thus, in view of Remark 4.1(i), the function X g satisfies the admissibility condition (26).
Recall that the classical continuous wavelet transform on [0, ∞[ is defined for suitable functions by where a > 0, b ≥ 0 and g ∈ L 2 ([0, ∞[, dx) is a classical wavelet on [0, ∞[, i.e., satisfying the admissibility condition A more complete and detailed discussion of the properties of the classical continuous wavelet transform on [0, ∞[ can be found in [14].
Remark 5.1 (i) According to [14], each function satisfying the conditions of Lemma 5.2 is a classical wavelet on [0, ∞[.
(ii) In view of (24), (26) and (37), g ∈ D(R) is a generalized wavelet, if and only if, t X g is a classical wavelet and we have The following statement provides a formula relating the generalized continuous wavelet transform to the classical one.

Lemma 5.3
Let g be as in Lemma 5.2. Then for all f ∈ L p α,n , p = 1 or 2, we have Proof. By (31) we have