Two Versions of Dunkl Linear Canonical Wavelet Transforms and Applications

Abstract

Among the class of generalized Fourier transformations, the linear canonical transform is of crucial importance, mainly due to its higher degrees of freedom compared to the conventional Fourier and fractional Fourier transforms. In this paper, we will introduce and study two versions of wavelet transforms associated with the linear canonical Dunkl transform. More precisely, we investigate some applications for Dunkl linear canonical wavelet transforms. Next we will introduce and develop the harmonic analysis associated with the Dunkl linear canonical wavelet packets transform. We introduce and study three types of wavelet packets along with their associated wavelet transforms. For each of these transforms, we establish a Plancherel and a reconstruction formula, and we analyze the associated scale-discrete scaling functions.

1. Introduction

The Fourier transform is regarded as one of the remarkable discoveries in mathematical science as it profoundly influenced diverse branches of science and engineering. In the realm of harmonic analysis, the Fourier transform plays a crucial role in analyzing signals wherein the characteristics are statistically invariant over time [1]. In the higher-dimensional scenario, there are several ways to arrive at the definition of the Fourier transform. The most basic formulation in ${\mathbb{R}}^N$ is given by the integral transform

$$\begin{array}{c}\hfill \mathcal{F}(f)(\lambda )=\frac{1}{{(2\pi )}^{N/2}}{\int }_{{\mathbb{R}}^N}f(x)e^{-i⟨\lambda ,x⟩}dx.\end{array}$$

(1)

Alternatively, one can rewrite the transform as

$$\begin{array}{c}\hfill \mathcal{F}(f)(\lambda )=\frac{1}{{(2\pi )}^{N/2}}{\int }_{{\mathbb{R}}^N}f(x)\mathcal{K}(\lambda ,x)dx,\end{array}$$

(2)

where $\mathcal{K}(·,·)$ is the unique solution to the system

$$\left\{\begin{array}{cc}{\partial }_{x_j}\mathcal{K}(\lambda ,x)=-i{\lambda }_j\mathcal{K}(\lambda ,x),\hfill & j=1,\dots ,N,\hfill \\ \mathcal{K}(\lambda ,0)=1,\hfill & \lambda \in {\mathbb{R}}^N.\hfill \end{array}\right.$$

Yet another mathematical description of the higher-dimensional Fourier transform was proposed by Howe [2] via the Laplace operator $\Delta$ on ${\mathbb{R}}^N$ as follows:

$$\begin{array}{c}\hfill \mathcal{F}=exp\left(\frac{i\pi N}{4}\right)exp\left(\frac{i\pi }{4}\left(\Delta -{|x|}^2\right)\right).\end{array}$$

(3)

It is pertinent to mention that each of the above alternative representations has its specific use cases, and a detailed description regarding different ramifications of the Fourier transform can be found in [3]. Many generalizations of the Fourier transform can be attributed to a deeper understanding of the fundamental operators in harmonic analysis. Recently, there has been a lot of interest in other differential or difference operators. The focus is in particular on the generalized Fourier transforms that subsequently arise from these operator theoretic notions, including the Dunkl transform [4], various discrete Fourier transforms in ${\mathbb{R}}^N$ [5], Fourier transforms in Clifford algebras [6], and many more.

Charles Dunkl [7] introduced and analyzed the Dunkl transform within the framework of extending the classical theory of spherical harmonics. To date, the Dunkl transform has been the subject of several studies in the field of harmonic analysis, including studies on the Bessel and Flett potentials [8], Babenko inequality [9], uncertainty principles [10,11,12], time frequency analysis and the Dunkl Gabor transform [13], real Paley–Wiener theorems [14], heat equations [15], Dunkl wavelet transforms [16], the generalized translation operator [17], the generalized maximal function [18], and many more.

This paper is a continuation of the works carried out in a previous article [19]. Nonetheless, in [20], we studied the harmonic analysis associated with the linear canonical Dunkl transform. More precisely, we introduced the generalized translation operators and the generalized convolution product associated with the LCDT and gave their main properties. Remember that the linear canonical transform (LCT) was first developed by Collins [21] in the field of paraxial optics, and also by Moshinsky–Quesne [22] in quantum mechanics. They used it to explore how information and uncertainty are preserved when phasespace is changed through linear transformations. The LCT is a type of mathematical transformation that works with a general linear mapping in phasespace that does not lose energy, and it has three adjustable parameters in total. The involved parameters constitute a $2\times 2$ uni-modular matrix mapping the position x and the wave number y into

$$\begin{array}{c}\hfill \begin{array}{c}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right),\hfill \end{array}\end{array}$$

where $ad-bc=1.$ The linear canonical transform of any signal f with respect to a real matrix $M=(a,b;c,d)$ that is in the group $SL(2,\mathbb{R})$, such that $b\ne 0$ is defined by

$$\begin{array}{c}\hfill {\mathcal{F}}^M(f)(y)=\frac{1}{\sqrt{ib}}{\int }_{\mathbb{R}}f(x){\mathcal{K}}^M(x,y)dy,\end{array}$$

(4)

where

$$\begin{array}{c}\hfill {\mathcal{K}}^M(x,y)=exp\left\{\frac{i}{2}\left(\frac{d{x}^2+a{y}^2-ixy}{b}\right)\right\}.\end{array}$$

(5)

It is important to note that the LCT offers a single approach to handle many types of generalized Fourier transforms. It includes several well-known integral transforms like the Fourier transform [1,23], the fractional Fourier transform [24], the Fresnel transform [25], scaling operations, and others [26,27]. Because it has more flexibility and a clear geometric meaning, the LCT is more adaptable than other transforms. This makes it a useful and strong tool for solving complex problems in fields such as optics, quantum physics, and signal processing [26,27,28]. In fact, over the past few decades, the use of the LCT has expanded quickly, and it is now being used for a wide range of problems including signal analysis, filter design, phase retrieval, pattern recognition, radar analysis, holographic three-dimensional television, and quantum physics, as well as in many other areas. Alongside its applications, the theoretical background of the LCT has also been widely explored. This has resulted in important mathematical results such as convolution theorems [29], sampling theorems [30], Poisson summation formulas [31], and uncertainty principles [32]. For more information on the LCT and its uses, see [26,27,28,33,34,35].

Among the class of generalized Fourier transformations, the linear canonical transform is of pivotal importance due mainly to its higher degrees of freedom compared to the conventional Fourier and fractional Fourier transforms. This article is motivated essentially by some interesting recent developments in the study of linear canonical Dunkl transforms (LCDTs) and their associated wavelet and wavelet packet transformations. This paper is a continuation of these recent works in the LCDT frame. We note that Ghazouani et al. have introduced this transform in [19]. Next, very recently, many authors have been investigating the behavior of the LCDT in several problems already studied for the usual Fourier transform, for instance, the translation operator and convolution product [20], wavelet transform [36,37], Gabor transform [38], uncertainty principles [39], Wigner transform [20], localization operators [20,36,38,40], wavelet multipliers [41], real Paley–Wiener theorems [42], and so on.

The aim in this paper is to presents a general construction of wavelet packets associated with the linear canonical Dunkl transform (LCDT). Specifically, we investigate three types of wavelet packets namely, the DLCP-wavelet packet, the DLC$\mathfrak{M}$-wavelet packet, and the DLCS-wavelet packet, along with their associated wavelet packet transforms. For each of these transforms, we establish both Plancherel and reconstruction formulas. In addition, we introduce two types of scale-discrete scaling functions related to the LCDT operator and derive their corresponding Plancherel and reconstruction results.

The remainder of this paper is arranged as follows: In Section 2 and Section 3, we recall the main results of the harmonic analysis associated with the Dunkl transform and the LCDT. Section 4 is devoted to the study of the generalized wavelet transform. In the final sections, we introduce three types of wavelet packets related to the LCDT and study the corresponding wavelet packet transforms. In particular, we derive the Plancherel and reconstruction formulas corresponding to these transforms, along with those associated with the discrete scale functions.

We close this section by summarizing in Table 1 the main symbols used in this paper.

Table 1. List of symbols.

Name	Symbol	Equations
Dunkl–Laplace operator	${\Delta }_k$	(10)
Dunkl kernel	$e_k$	(11)
Dunkl transform	${\mathfrak{F}}_D$	(17)
Dunkl translation	${\tau }_t$	(25)
Dunkl convolution	${\ast }_D$	(36)
Linear canonical Dunkl transform (LCDT)	${\mathfrak{F}}_D^M$	(41)
LCD kernel	${\mathfrak{B}}_k^M$	(42)
Dunkl–Laplace operator	${▵}_k^M$	(43)
Generalized translation operator	$T_x^M$	(60)
Generalized convolution product	$\underset{M}{\ast }$	(69)
Admissible Linear canonical wavelet constant	$C_{\phi }^M$	(76)
Linear canonical Dunkl wavelet transform	${\mathfrak{W}}_{\phi }^M$	(82)
DLC P-wavelet packet transform	${\Psi }_g^{p,M}$	(111)
DLC modified wavelet packet transform	${\Psi }_g^{p,M}$	(130)
DLC S-wavelet packet transform	${\Psi }_g^{s,M}$	(138)

2. Linear Canonical Dunkl Transform

In this section, we will recall the prerequisites for the Dunkl and the linear canonical Dunkl transforms [4,7,16,19,20,43,44].

2.1. Dunkl Operators

The reflection in the hyperplane $H_s\subset {\mathbb{R}}^N$, which is orthogonal to a nonzero $s\in {\mathbb{R}}^N$, is expressed by

$${\sigma }_s(t)=t-2\frac{⟨s,t⟩}{{|s|}^2}s,$$

(6)

where $|s|=\sqrt{⟨s,s⟩}$ is the usual norm on ${\mathbb{R}}^N$. Then a finite subset $\mathcal{R}$ of ${\mathbb{R}}^N\setminus \left\{0\right\}$ is called a root system, if for every $s\in \mathcal{R}$, ${\sigma }_s(\mathcal{R})=\mathcal{R}$, and $\mathcal{R}\cap \mathbb{R}s=\{\pm s\}$. Therefore, all reflections ${\sigma }_s,s\in \mathcal{R}$ generate a finite group W, which is a subset of $O(N)$. This group is commonly referred to as the reflection group. Moreover, a positive function k defined on $\mathcal{R}$ is said to be a multiplicity function, if it is invariant under the action W.

For $t\in {\mathbb{R}}^N\setminus {\bigcup }_{s\in \mathcal{R}}{H}_s$, let ${\mathcal{R}}_{+}=\left\{s\in \mathcal{R}:⟨s,t⟩>0\right\}$ be a positive root system, such that ${|s|}^2=2$, for every $s\in {\mathcal{R}}_{+}$. Then we introduce ${\omega }_k$ by

$${\omega }_k(t)=\prod _{s\in {\mathcal{R}}_{+}}{|⟨s,t⟩|}^{2k(s)},$$

(7)

and we define

$$⟨k⟩=\sum _{s\in {\mathcal{R}}_{+}}k(s)\mathrm{and}{c}_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{{|t|}^2}{2}}{\omega }_k(t)dt.$$

(8)

For any orthonormal basis ${\{e_j\}}_{j=1}^N$ of ${\mathbb{R}}^N$, the Dunkl operators $T_j$, are defined by

$$T_{j}f(t):=\frac{\partial f}{\partial t_j}(t)+\sum _{s\in {\mathcal{R}}_{+}}k(s)s_j\frac{f(t)-f({\sigma }_s(t))}{⟨s,t⟩},t\in {\mathbb{R}}_{\mathrm{reg}}^N={\mathbb{R}}^N\setminus {\cup }_{s\in \mathcal{R}}{H}_s,$$

(9)

where $s_j=⟨s,e_j⟩$. Then the Dunkl–Laplacian operator ${\Delta }_k$ is defined by

$${\Delta }_{k}f(t)={\displaystyle {\displaystyle \sum _{j=1}^N}}{T}_j^{2}f(t)=\Delta f(t)+2\sum _{s\in {\mathcal{R}}^{+}}k(s)\left(\frac{⟨\nabla f(t),s⟩}{⟨s,t⟩}-\frac{f(t)-f({\sigma }_s(t))}{{⟨s,t⟩}^2}\right),t\in {\mathbb{R}}_{\mathrm{reg}}^N,$$

(10)

where $\Delta$ is the Euclidean Laplacian and ∇ is the gradient operator on ${\mathbb{R}}^N$.

Let $x\in {\mathbb{R}}^N$. Then the system

$$\left\{\begin{array}{ccc}{T}_{j}v(t,x)\hfill & =\hfill & y_{j}v(t,x),j=1,\dots ,N,\hfill \\ v(0,x)\hfill & =\hfill & 1,\hfill \end{array}\right.$$

(11)

has one special solution $e_k(.,y)$, known as the Dunkl kernel. This solution is an analytic on ${\mathbb{R}}^N$ that has a unique holomorphic extension to ${\mathbb{C}}^N\times {\mathbb{C}}^N$ and satisfies the following conditions:

• For every $\lambda \in \mathbb{C}$ and $t,x\in {\mathbb{C}}^N$,

  $$e_k(x,t)=e_k(t,x),e_k(x,0)=1,e_k(\lambda x,t)=e_k(x,\lambda t),$$

  and we have [15],

  $$e_k(t,x)={\int }_{{\mathbb{R}}^N}exp(⟨z,x⟩)d{\nu }_t(z),$$

  (12)

  where ${\nu }_t$ represents the positive probability measure on ${\mathbb{R}}^N$ supported on the ball $B_N(0,|t|)$.
• For every $t\in {\mathbb{R}}^N$, $x\in {\mathbb{C}}^N$, and $n\in {\mathbb{N}}^N,$

  $$|D_x^{\nu }{e}_k(t,x)|\le {|t|}^{|n|}exp\left(|t||\mathrm{Re}(x)|\right),$$

  (13)

  where $D_z^n=\frac{{\partial }^{|n|}}{\partial x_1^{{\nu }_1}\cdots \partial x_N^{{\nu }_N}}$ and $|n|=n_1+\cdots +n_N$. Especially,

  $$\forall t,x\in {\mathbb{R}}^N,|e_k(it,x)|\le 1.$$

  (14)

For $p\in [1,\infty ]$, we denote by $p^{\prime }$ its conjugate exponent, and by $L_k^p({\mathbb{R}}^N)$ the space of measurable functions u on ${\mathbb{R}}^N$, such that for $p\ge 1$,

$${\parallel u\parallel }_{L_k^p}={\left({\int }_{{\mathbb{R}}^N}{|u(x)|}^p{\gamma }_k(dx)\right)}^{1/p}<\infty ,$$

where ${\gamma }_k(dx):={c_k}^{-1}{\omega }_k(x)dx$, and for $p=\infty$,

$${\parallel u\parallel }_{L_k^{\infty }}=\mathrm{ess}\underset{x\in {\mathbb{R}}^N}{sup}|u(x)|<\infty .$$

Then for any radial function $u\in L_k^1({\mathbb{R}}^N)$, we have [44]

$${\int }_{{\mathbb{R}}^N}u(t){\gamma }_k(dt)=d_k{\int }_{{\mathbb{R}}^N}v(r)r^{2⟨k⟩+N-1}dr,$$

(15)

where $u(·)=v(|·|)$ and

$$d_k:={\left(2^{N/2+⟨k⟩-1}\Gamma (N/2+⟨k⟩)\right)}^{-1}.$$

(16)

The Dunkl transform is defined on $L_k^1({\mathbb{R}}^N)$ by

$${\mathfrak{F}}_D(u)(\xi )={\int }_{{\mathbb{R}}^N}u(t)e_k(-ix,\xi ){\gamma }_k(dx),$$

(17)

with

$$\parallel {\mathfrak{F}}_D{(u)\parallel }_{L_k^{\infty }}\le {\parallel u\parallel }_{L_k^1}.$$

(18)

Moreover, if $u\in L_k^1({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D(u)$ belongs to $L_k^1({\mathbb{R}}^N)$, we have the following inversion formula:

$${\mathfrak{F}}_D^{-1}(u)(·)={\mathfrak{F}}_D(u)(-·),a.e.$$

(19)

The Dunkl transform is a topological isomorphism from the Schwartz space $\mathcal{S}({\mathbb{R}}^N)$ onto itself and satisfies the following conditions [7,43]:

• For any $u\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D^{-1}(u)(x)={\mathfrak{F}}_D(u)(-x),x\in {\mathbb{R}}^N.$$

  (20)
• For every $u\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D(\overline{u})=\overline{{\mathfrak{F}}_D(\breve{u})},$$

  (21)

  and

  $${\mathfrak{F}}_D(u)(·)={\mathfrak{F}}_D(\breve{u})(-·),$$

  (22)

  where $\breve{u}(t)=u(-t)$.
• Parseval-type relation: For all $u,v\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\int }_{{\mathbb{R}}^N}u(t)\overline{v(t)}{\gamma }_k(dt)={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D(u)(\lambda )\overline{{\mathfrak{F}}_D(v)(\lambda )}{\gamma }_k(d\lambda ).$$

  (23)
• Plancherel-type relation: For every $u\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\int }_{{\mathbb{R}}^N}{|u(t)|}^2{\gamma }_k(dt)={\int }_{{\mathbb{R}}^N}{|{\mathfrak{F}}_D(u)(\lambda )|}^2{\gamma }_k(d\lambda ).$$

  (24)

Definition 1. 

Let $t\in {\mathbb{R}}^N$. The Dunkl translation operator ${\tau }_t$ is defined on $L_k^2({\mathbb{R}}^N)$ by [44]

$${\mathfrak{F}}_D({\tau }_{t}u)=e_k(it,·){\mathfrak{F}}_D(u).$$

(25)

Let ${\mathcal{W}}_k({\mathbb{R}}^N):=\left\{u\in L_k^1({\mathbb{R}}^N):{\mathfrak{F}}_D(u)\in L_k^1({\mathbb{R}}^N)\right\}$ be the space of all functions u satisfying (25) point-wise. Then we have the following [17,18]:

• For every $u\in L_k^2({\mathbb{R}}^N)$,

  $$\parallel {\tau }_t{u\parallel }_{L_k^2}\le {\parallel u\parallel }_{L_k^2}.$$

  (26)
• For every $u\in {\mathcal{W}}_k({\mathbb{R}}^N)$,

  $${\tau }_{t}u(y)={\int }_{{\mathbb{R}}^N}{e}_k(it,\lambda )e_k(iy,\lambda ){\mathfrak{F}}_D(u)(\lambda ){\gamma }_k(d\lambda ),y\in {\mathbb{R}}^N.$$

  (27)
• For every $u\in {\mathcal{W}}_k({\mathbb{R}}^N)$,

  $${\tau }_{t}u(y)={\tau }_y(u)(t),y\in {\mathbb{R}}^N.$$

  (28)
• The Dunkl translation operator is also well defined (see [18,45]) on $L_k^p({\mathbb{R}}^N)$, $1\le p\le \infty$ when $W={\mathbb{Z}}_2^N$, $C^{\infty }({\mathbb{R}}^N)$, and $L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$, $1\le p\le \infty$, the subspace of radial functions in $L_k^p({\mathbb{R}}^N)$.

Up until now, the Dunkl translation operator has only been explicitly defined when $N=1$ and $W={\mathbb{Z}}_2$. More precisely, for every $y\in \mathbb{R}$ and $u\in C(\mathbb{R})$,

$$\begin{array}{ccc}\hfill {\tau }_{t}u(y)& =& \frac{1}{2}{\int }_{-1}^{1}u\left(\sqrt{t^2+y^2-2xyt}\right)\left(1+\frac{t-y}{\sqrt{y^2+t^2-2txy}}\right){\Phi }_k(x){\gamma }_k(dx)\hfill \\ & +& \frac{1}{2}{\int }_{-1}^{1}u\left(-\sqrt{y^2+t^2-2txy}\right)\left(1-\frac{t-y}{\sqrt{y^2+t^2-2txy}}\right){\Phi }_k(x){\gamma }_k(dx),\hfill \end{array}$$

where

$${\Phi }_k(x)=\frac{\Gamma (k+\frac{1}{2})}{\sqrt{\pi }\Gamma (k)}(1+x){(1-x^2)}^{k-1}.$$

Consequently, in the case of $W={\mathbb{Z}}_2^N$, the author in [16] has determined a formulation for the Dunkl translation operator; that is, there exists a measure $d{\zeta }_{x,y}^k$ such that

$${\tau }_{x}u(y)={\int }_{\mathbb{R}}u(z)d{\zeta }_{x,y}^k(z),u\in C_b({\mathbb{R}}^N),$$

(29)

where

$$d{\zeta }_{x,y}^k(z)=\left\{\begin{array}{ccc}{\mathcal{I}}_k(x,y,z){\gamma }_k(dz),\hfill & \mathrm{if}\hfill & x,y\ne 0,\hfill \\ d{\delta }_x(z),\hfill & \mathrm{if}\hfill & y=0,\hfill \\ d{\delta }_y(z),\hfill & \mathrm{if}\hfill & x=0,\hfill \end{array}\right.$$

(30)

and ${\mathcal{I}}_k(x,y,z)$ is explicitly defined, in [16]. Moreover we have the following results [18].

Proposition 1. 

Let $1\le p\le \infty$ and $t\in {\mathbb{R}}^N$.

1. 

If $W={\mathbb{Z}}_2^N$, then for every $u\in L_k^p({\mathbb{R}}^N)$,

$$\parallel {\tau }_t{u\parallel }_{L_k^2}\le 2^{\frac{N}{2p}|2-p|}{\parallel u\parallel }_{L_k^2}.$$

(31)

In addition, if u is radial and belongs to ${\mathcal{W}}_k({\mathbb{R}}^N)$, then

$${\tau }_{t}u(x)={\int }_{B_N(0,|x|)}{u}_0(\sqrt{{|t|}^2+{|x|}^2+2⟨x,y⟩})d{\nu }_x(y),$$

(32)

where $u(y)=u_0(|y|)$.

2. 

If $u\in L_{k,\mathrm{rad}}^1({\mathbb{R}}^N)$ is positive, then

$${\tau }_{t}u\ge 0,{\tau }_{t}u\in L_k^1({\mathbb{R}}^N),$$

(33)

and

$${\int }_{{\mathbb{R}}^N}{\tau }_{y}u(t){\gamma }_k(dt)={\int }_{{\mathbb{R}}^N}u(x){\gamma }_k(dx).$$

(34)

3. 

For every $u\in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$,

$$\parallel {\tau }_t{u\parallel }_{L_k^2}\le {\parallel u\parallel }_{L_k^2}.$$

(35)

Definition 2. 

The Dunkl convolution product of any two functions $u,v$ is given by [18,45]

$$u{\ast }_{D}v(t)={\int }_{{\mathbb{R}}^N}{\tau }_{t}u(-x)v(x){\gamma }_k(dx),t\in {\mathbb{R}}^N.$$

(36)

This operator is both associative and commutative, and fulfils the following relations [18,45].

Proposition 2. 

Let $1\le p,q,r\le \infty ,$ such that $\frac{1}{p}+\frac{1}{q}-\frac{1}{r}=1.$

1. 

If $u\in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$ and $v\in L_k^q({\mathbb{R}}^N),$ then $u{\ast }_{D}v$ belongs to $L_k^r({\mathbb{R}}^N)$ and

$${∥u{\ast }_{D}v∥}_{L_k^r}\le {∥u∥}_{L_k^p}{∥v∥}_{L_k^q}.$$

(37)

2. 

In the case of $W={\mathbb{Z}}_2^N$, we have for all $v\in L_k^q({\mathbb{R}}^N)$ and $u\in L_k^p({\mathbb{R}}^N)$, the function $u{\ast }_{D}v\in L_k^r({\mathbb{R}}^N)$, such that

$${∥u{\ast }_{D}v∥}_{L_k^r}\le 2^{\frac{N}{2p}|p-2|}{∥u∥}_{L_k^p}{∥v∥}_{L_k^q}.$$

(38)

3. 

If $u,v\in L_k^2({\mathbb{R}}^N)$, then $u{\ast }_{D}v\in L_k^2({\mathbb{R}}^N)$ if and only if ${\mathfrak{F}}_D(u){\mathfrak{F}}_D(v)\in L_k^2({\mathbb{R}}^N)$, and

$${\mathfrak{F}}_D(u{\ast }_{D}v)={\mathfrak{F}}_D(u){\mathfrak{F}}_D(v).$$

(39)

4. 

If $u,v\in L_k^2({\mathbb{R}}^N)$, then

$${\int }_{{\mathbb{R}}^N}|u{\ast }_D{v(t)|}^2{\gamma }_k(dt)={\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D{(u)(\lambda )|}^2{|{\mathfrak{F}}_D(v)(\lambda )|}^2{\gamma }_k(d\lambda ).$$

(40)

2.2. LCDT

Throughout this paper $M:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is a matrix such that $ad-cb=1$ and $b\ne 0.$ This subsection reviews several results established in previous works [19,20].

Definition 3. 

The LCDT of a function $u\in L_k^1({\mathbb{R}}^N)$ is defined by

$$\forall t\in {\mathbb{R}}^N,{\mathfrak{F}}_D^M(u)(t)=\frac{1}{{(ib)}^{N/2+⟨k⟩}}{\int }_{{\mathbb{R}}^N}{\mathfrak{B}}_k^M(t,s)u(s){\gamma }_k(ds),$$

(41)

where

$${\mathfrak{B}}_k^M(t,s)=e^{\frac{i}{2}\left(\frac{d}{b}{|t|}^2+\frac{a}{b}{|s|}^2\right)}{e}_k(-it/b,s).$$

(42)

Let ${\Delta }_k^M$ be the operator defined by

$${\Delta }_k^M:={\Delta }_k-i\frac{d}{b}\left({\displaystyle \sum _{j=1}^N}{P}_j{T}_j-T_j{P}_j\right)-\frac{d^2}{b^2}{|t|}^2,$$

(43)

where $P_j(u(t)):=t_{j}u(t)$:

• We have

  $$e^{-\frac{i}{2}\frac{d}{b}{|t|}^2}\circ {\Delta }_k^M\circ e^{\frac{i}{2}\frac{d}{b}{|t|}^2}={\Delta }_k.$$

  (44)
• For every $u,v\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\int }_{{\mathbb{R}}^N}{\Delta }_k^{M}u(x)\overline{v(x)}{\gamma }_k(dx)={\int }_{{\mathbb{R}}^N}u(x)\overline{{\Delta }_k^{M}v(x)}{\gamma }_k(dx).$$

  (45)
• For every $s\in {\mathbb{R}}^N$, ${\mathfrak{B}}_k^M(·,s)$ satisfies

  $$\left\{\begin{array}{c}{\Delta }_k^M{\mathfrak{B}}_k^M(·,s)=-{|\frac{s}{b}|}^2{\mathfrak{B}}_k^M(·,s),\hfill \\ \\ {\mathfrak{B}}_k^M(0,s)=e^{\frac{i}{2}\frac{a}{b}{|s|}^2}.\hfill \end{array}\right.$$

  (46)
• For all $u\in \mathcal{S}({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D^M\left({|s|}^{2}u(s)\right)=-b^2{\Delta }_k^M\left({\mathfrak{F}}_D^M(u)\right),$$

  (47)

  and

  $${|t|}^2{\mathfrak{F}}_D^M(u)=-b^2{\mathfrak{F}}_D^M\left({\Delta }_k^{M^{-1}}u\right).$$

  (48)
• For each $t,s\in {\mathbb{R}}^N$,

  $$|{\mathfrak{B}}_k^M(t,s)|\le 1.$$

  (49)

Notice that [19] if $M:=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, then the LCDT is the Dunkl transform, if $M=\left(\begin{array}{cc}1& \tau \\ 0& 1\end{array}\right)$, then the LCDT is the Fresnel–Dunkl transform, and in the case of $M=\left(\begin{array}{cc}cos(\alpha )& sin(\alpha )\\ -sin(\alpha )& cos(\alpha )\end{array}\right)$, the LCDT is with the Dunkl-fractional transform (see [19] for more examples).

2.2.1. LCDT on $L_k^p({\mathbb{R}}^N)$, $1\le p\le 2$

For $s\in \mathbb{R}$ we define the operators ${\mathbf{L}}_s$ and ${\delta }_s$ by

$${\mathbf{L}}_{s}f(t)=e^{\frac{is}{2}{|t|}^2}f(t)\mathrm{and}{\delta }_{s}f(t)=\frac{1}{{|s|}^{⟨k⟩+N/2}}f(t/s),s\ne 0.$$

(50)

Then we have the following results on $L_k^1({\mathbb{R}}^N)$:

• For all $s\ne 0$,

  $${\delta }_s\circ {\mathfrak{F}}_D={\mathfrak{F}}_D\circ {\delta }_{\frac{1}{s}}.$$

  (51)
• We have

  $$e^{i(⟨k⟩+N/2)\frac{\pi }{2}sgn(b)}{\mathfrak{F}}_D^M={\mathbf{L}}_{\frac{d}{b}}\circ {\delta }_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{\frac{a}{b}}.$$

  (52)
• The LCDT belongs to $C_0({\mathbb{R}}^N)$, such that

  $$\parallel {\mathfrak{F}}_D^M{(u)\parallel }_{L_k^{\infty }}\le {|b|}^{-⟨k⟩+N/2}{\parallel u\parallel }_{L_k^1}.$$

  (53)

Theorem 1. 

1. 

For all $u,v\in L_k^1({\mathbb{R}}^N)$,

$${\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(u)(\lambda )\overline{v(\lambda )}{\gamma }_k(d\lambda )={\int }_{{\mathbb{R}}^N}u(t)\overline{{\mathfrak{F}}_D^{M^{-1}}(v)(t)}{\gamma }_k(dt),$$

(54)

where $M^{-1}$ is the inverse of M.

2. 

Plancherel-type formula: If $u\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$, then ${\mathfrak{F}}_D^M(u)\in L_k^2({\mathbb{R}}^N)$, such that

$$\parallel {\mathfrak{F}}_D^M{(u)\parallel }_{L_k^2}={\parallel u\parallel }_{L_k^2}.$$

(55)

3. 

Parseval-type formula: For all $u,v\in L_k^2({\mathbb{R}}^N)$,

$${⟨{\mathfrak{F}}_D^M(u),v⟩}_{L_k^2}={⟨u,{\mathfrak{F}}_D^{M^{-1}}v⟩}_{L_k^2}.$$

(56)

4. 

Inversion formula: For all $u\in L_k^1({\mathbb{R}}^N)$ with ${\mathfrak{F}}_D^M(u)\in L_k^1({\mathbb{R}}^N),$

$$\left({\mathfrak{F}}_D^M\circ {\mathfrak{F}}_D^{M^{-1}}\right)(u)=\left({\mathfrak{F}}_D^{M^{-1}}\circ {\mathfrak{F}}_D^M\right)(u)=u,a.e.$$

(57)

Definition 4. 

For $1\le p\le 2$, we define the LCDT on $L_k^p({\mathbb{R}}^N)$ by

$${\mathfrak{F}}_D^M=e^{-i(N/2+⟨k⟩)\frac{\pi }{2}\mathrm{sgn}(b)}\left({\mathbf{L}}_{\frac{d}{b}}\circ {\delta }_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{\frac{a}{b}}\right).$$

(58)

Then we have the following Young-type inequality,

$$\parallel {\mathfrak{F}}_D^M(u){\parallel }_{L_k^{p^{\prime }}}\le {|b|}^{\left(⟨k⟩+N/2\right)\left(2/p^{\prime }-1\right)}{\parallel u\parallel }_{L_k^p}.$$

(59)

2.2.2. Generalized Convolution Product

We define the generalized translation operator associated with ${\Delta }_k^M$ by [20],

$$T_x^{M}u(y)=e^{\frac{i}{2}\frac{d}{b}{(|x|}^2+{|y|}^2)}{\tau }_x\left(e^{-\frac{i}{2}\frac{d}{b}{|s|}^2}u(s)\right)(y),x,y\in {\mathbb{R}}^N.$$

(60)

Equation (60) is valid for functions on the spaces $L_k^2({\mathbb{R}}^N)$, $\mathcal{S}({\mathbb{R}}^N)$, $L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$, $1\le p\le \infty$, $C^{\infty }({\mathbb{R}}^N)$, and $L_k^p({\mathbb{R}}^N)$, $1\le p\le \infty$, when $W={\mathbb{Z}}_2^N$. Then we have following relations:

• $T_0^M=\mathrm{Id}$ and $T_x^{M}f(y)=T_y^{M}f(x).$
• For every $x,y,z\in {\mathbb{R}}^N,$

  $$\begin{array}{c}{T}_x^M\left[{\mathfrak{B}}_k^M(.,y)\right](z)=e^{-\frac{i}{2}\frac{a}{b}{|y|}^2}{\mathfrak{B}}_k^M(x,y){\mathfrak{B}}_k^M(z,y).\hfill \end{array}$$

  (61)
• The operator $T_x^M$ is continuous from $C_{b,\mathrm{rad}}({\mathbb{R}}^N)$ onto $C_b({\mathbb{R}}^N)$, $\mathcal{S}({\mathbb{R}}^N)$ onto itself, $L_k^2({\mathbb{R}}^N)$ onto itself, and on $L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$, such that, for every $u\in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N),$$1\le p\le \infty$,

  $${∥T_x^{M}u∥}_{L_k^p}\le {∥u∥}_{L_k^p},$$

  (62)

  and for every $u\in L_k^2({\mathbb{R}}^N)$,

  $${∥T_x^{M}u∥}_{L_k^2}\le {∥u∥}_{L_k^2}.$$

  (63)
• In the case of $W={\mathbb{Z}}_2^N$, for every $u\in L_k^p({\mathbb{R}}^N)$, $1\le p\le \infty$,

  $${∥T_x^{M}u∥}_{L_k^p}\le 2^{\frac{N}{2p}|p-2|}{∥u∥}_{L_k^p}.$$

  (64)
• For every $u\in L_{k,\mathrm{rad}}^1({\mathbb{R}}^N)$ or $\mathcal{S}({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D^M\left[T_x^{M^{-1}}u\right](y)=e^{\frac{i}{2}\frac{d}{b}{|y|}^2}\overline{{\mathfrak{B}}_k^M(y,x)}{\mathfrak{F}}_D^M(u)(y).$$

  (65)
• For every $u\in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N),p\in (1,2],$

  $${\mathfrak{F}}_D^M\left[T_x^{M^{-1}}u\right](y)=e^{\frac{i}{2}\frac{d}{b}{|y|}^2}\overline{{\mathfrak{B}}_k^M(y,x)}{\mathfrak{F}}_D^M(u)(y),a.e.$$

  (66)
• For every $x,y\in {\mathbb{R}}^N$,

  $$T_x^{M}u(y)={\int }_{{\mathbb{R}}^N}{\mathrm{e}}^{-i\frac{d}{b}{z}^2}u(z){\mathcal{I}}_k^M(x,y,z){\gamma }_k(dz),$$

  (67)

  where ${\mathcal{I}}_k^M(x,y,z)={\mathrm{e}}^{\frac{i}{2}\frac{d}{b}\left({|x|}^2+{|y|}^2+{|z|}^2\right)}{\mathcal{I}}_k(x,y,z)$.
• For every $u\in \mathcal{S}({\mathbb{R}}^N)$,

  $$T_x^{M^{-1}}u(t)={(-ib)}^{-⟨k⟩+N/2}{e}^{-\frac{i}{2}\frac{a}{b}{|t|}^2}{\int }_{{\mathbb{R}}^N}{e}_k(i\xi /b,t)\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\mathfrak{F}}_D^M(u)(\xi ){\gamma }_k(d\xi ).$$

  (68)

The generalized convolution product operator in the LCDT setting is then defined by

$$u\underset{M}{\ast }v(t)={\int }_{{\mathbb{R}}^N}{T}_t^{M}u(-x)e^{-i\frac{d}{b}{|x|}^2}v(x){\gamma }_k(dy).$$

(69)

Then we have the following:

• $v\underset{M}{\ast }u=u\underset{M}{\ast }v.$
• $u\underset{M}{\ast }\left(T_x^{M}v\right)=\left(T_x^{M}u\right)\underset{M}{\ast }v=T_x^M\left(u\underset{M}{\ast }v\right).$

Proposition 3 

(Young’s Inequality). In the case of $W={\mathbb{Z}}_2^N$, if $u\in L_k^p({\mathbb{R}}^N)$ and $v\in L_k^q({\mathbb{R}}^N)$, then $u\underset{M}{\ast }v\in L_k^r({\mathbb{R}}^N)$, such that

$${∥u\underset{M}{\ast }v∥}_{L_k^r}\le 2^{\frac{N}{2p}|p-2|}{∥u∥}_{L_k^p}{∥v∥}_{L_k^q},$$

(70)

where $p,q,r\in [1,\infty ]$ are such that $p^{-1}+q^{-1}=r^{-1}+1.$ In particular, if $u\in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$ and $v\in L_k^q({\mathbb{R}}^N),$ then

$${∥u\underset{M}{\ast }v∥}_{L_k^r}\le {∥u∥}_{L_k^p}{∥v∥}_{L_k^q}.$$

(71)

Moreover, the generalized convolution product operator satisfies the following relations:

• If $v\in L_k^2({\mathbb{R}}^N)$ and $u\in L_k^1({\mathbb{R}}^N)$, then

  $${\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{\ast }v\right)(\xi )={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{i}{2}\frac{d}{b}{|\xi |}^2}{\mathfrak{F}}_D^M(u)(\xi ){\mathfrak{F}}_D^M(v)(\xi ),a.e.$$

  (72)
• If $v\in L_k^1({\mathbb{R}}^N)$ and $u\in L_{k,\mathrm{rad}}^1({\mathbb{R}}^N)$, then

  $${\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{\ast }v\right)(\xi )={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{i}{2}\frac{d}{b}{|\xi |}^2}{\mathfrak{F}}_D^M(u)(\xi ){\mathfrak{F}}_D^M(v)(\xi ).$$

  (73)
• If $v\in L_k^p({\mathbb{R}}^N),p\in (1,2]$ and $u\in L_{k,\mathrm{rad}}^1({\mathbb{R}}^N)$, then

  $${\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{\ast }v\right)(\xi )={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{i}{2}\frac{d}{b}{|\xi |}^2}{\mathfrak{F}}_D^M(u)(\xi ){\mathfrak{F}}_D^M(v)(\xi ),\mathrm{a.e.}$$

  (74)
• If $u,v,w\in L_{k,\mathrm{rad}}^1({\mathbb{R}}^N)$, then

  $$\left(u\underset{M}{\ast }v\right)u\underset{M}{\ast }\left(v\underset{M}{\ast }w\right)=\underset{M}{\ast }w.$$
• If $u,v\in L_k^2({\mathbb{R}}^N),$ then

  $${\int }_{{\mathbb{R}}^N}{\left|\left(u\underset{M^{-1}}{\ast }v\right)(t)\right|}^2{\gamma }_k(dt)={|b|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M{(u)(\xi )|}^2{|{\mathfrak{F}}_D^M(v)(\xi )|}^2{\gamma }_k(d\xi ).$$

  (75)

3. The DLCWT

Let ${\mathbb{R}}_{+}^{N+1}:=\{(t,r)\in {\mathbb{R}}^{N+1}:r>0\}$, and for $1⩽p⩽\infty$, we denote by $L_{{\mu }_k}^p\left({\mathbb{R}}_{+}^{N+1}\right)$ the space of measurable functions f on ${\mathbb{R}}_{+}^{N+1}$, such that

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_{{\mu }_k}^p}& :=& {\left({\int }_{{\mathbb{R}}_{+}^{N+1}}{|f(t,r)|}^{p}d{\mu }_k(t,r)\right)}^{1/p}<\infty ,\hfill \\ \hfill {\parallel f\parallel }_{L_{{\mu }_k}^{\infty }}& :=& \begin{array}{c}esssup\\ (t,r)\in {\mathbb{R}}_{+}^{N+1}\end{array}|f(t,r)|<\infty ,\hfill \end{array}$$

where ${\mu }_k$ is weight measure given by $d{\mu }_k(t,r)={\gamma }_k(dt)r^{-(2⟨k⟩+N+1)}dr$.

Definition 5. 

Let $\phi \in L_k^2({\mathbb{R}}^N)$. We say that φ is an admissible Dunkl linear canonical wavelet (ADLCW) if, for almost all $y\in {\mathbb{R}}^N$,

$$0<C_{\phi }^M:={\int }_0^{\infty }|{\mathfrak{F}}_D^M(\phi )(ty){|}^2\frac{dt}{t}<\infty .$$

(76)

The primary motivation for using an ADLCW is to ensure that the generalized wavelet transform is associated with the LCDT, an investigation shown in this section that can be inverted, allowing a perfect reconstruction of the original signal from its wavelet representation. Without admissibility, the generalized wavelet transform might lose information, making it unsuitable for applications requiring accurate signal representation and analysis.

Let $r>0$, $t\in {\mathbb{R}}^N$ and $\phi \in L_k^2({\mathbb{R}}^N)$. We define the function ${\phi }_{r,t}^M$ on ${\mathbb{R}}^N$ by

$${\phi }_{r,t}^M(y)=r^{⟨k⟩+N/2}\overline{T_t^{M^{-1}}({\phi }_r^M)(y)},$$

(77)

where

$${\phi }_r^M(y):=r^{-2⟨k⟩-N}{e}^{-\frac{i}{2}\frac{a}{b}(1-1/r^2){|y|}^2}\phi (y/r).$$

(78)

Then, we have the following:

• For $\phi \in L_k^2({\mathbb{R}}^N)$,

  $${\parallel {\phi }_{r,t}^M\parallel }_{L_k^2}\le {\parallel \phi \parallel }_{L_k^2}.$$

  (79)
• For $\phi \in L_k^1({\mathbb{R}}^N)\cup L_k^2({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D^M({\phi }_r^M)(\xi ):=e^{\frac{i}{2}\frac{d}{b}(1-r^2){|\xi |}^2}{\mathfrak{F}}_D^M(\phi )(r\xi ).$$

  (80)
• For $\phi \in L_k^1({\mathbb{R}}^N)\cup L_k^2({\mathbb{R}}^N)$,

  $${\mathfrak{F}}_D^M(\overline{{\phi }_{r,t}^M})(\xi ):=r^{⟨k⟩+N/2}{e}^{\frac{i}{2}\frac{d}{b}(2-r^2){|\xi |}^2}\overline{{\mathfrak{B}}_k^M(\xi ,t)}{\mathfrak{F}}_D^M(\phi )(r\xi ).$$

  (81)

Finally, we are in a position to present the formal definition of the Dunkl linear canonical wavelet transform.

Definition 6. 

The Dunkl linear canonical wavelet transform (DLCWT) of any functions $f\in L_k^2({\mathbb{R}}^N)$ is denoted by ${\mathfrak{W}}_{\phi }^M(f)$ and is defined as

$${\mathfrak{W}}_{\phi }^M(f)(t,r):={\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{|y|}^2}f(y)\overline{{\phi }_{r,t}^M(-y)}{\gamma }_k(dy)=r^{⟨k⟩+N/2}f\underset{M^{-1}}{\ast }{\phi }_r^M(t),(t,r)\in {\mathbb{R}}_{+}^{N+1},$$

(82)

where ${\phi }_{r,t}^M$ is given by (77).

It satisfies the following properties:

• For any $f\in L_k^2({\mathbb{R}}^N)$,

  $${\parallel {\mathfrak{W}}_{\phi }^M(f)\parallel }_{L_{{\mu }_k}^{\infty }}\le {\parallel f\parallel }_{L_k^2}{\parallel \phi \parallel }_{L_k^2}.$$

  (83)
• If $\phi \in L_{k,\mathrm{rad}}^p({\mathbb{R}}^N)$, $p\in [1,\infty ]$, then for any $f\in L_k^{p^{\prime }}({\mathbb{R}}^N)$

  $$|{\mathfrak{W}}_{\phi }^{M}f(t,r)|\le r^{(2⟨k⟩+N)(\frac{1}{p}-\frac{1}{2})}{\parallel f\parallel }_{L_k^{p^{\prime }}}{\parallel \phi \parallel }_{L_k^p}.$$

  (84)
• Let $\phi \in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ and $f\in L_k^2({\mathbb{R}}^N)$. We have

  $${\mathfrak{F}}_D^M\left({\mathfrak{W}}_{\phi }^M(f)(r,.)\right)(\lambda )={(ibr)}^{⟨k⟩+N/2}{e}^{-\frac{i}{2}\frac{d}{b}{|\lambda |}^2}{\mathfrak{F}}_D^M(f)(\lambda ){\mathfrak{F}}_D^M({\phi }_r^M)(\lambda ).$$

  (85)
• Let $f,\phi \in L_k^2({\mathbb{R}}^N)$, for any $\lambda >0$ and $(t,r)\in {\mathbb{R}}_{+}^{N+1}$

  $${\mathfrak{W}}_{\phi }^M(f_{\lambda }^M)(t,r)=e^{-i\frac{a}{2b}\frac{{\lambda }^2-1}{{\lambda }^2}{|t|}^2}{\mathfrak{W}}_{\phi }^M(f)\left(\frac{t}{\lambda },\frac{r}{\lambda }\right).$$

  (86)

Theorem 2 

(Plancherel-type formula). Let φ be an ADLCW. Then for all $f\in L_k^2({\mathbb{R}}^N)$, we have

$${|b|}^{-2⟨k⟩-N}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{W}}_{\phi }^M(f)(t,r)\right|}^{2}d{\mu }_k(t,r)=C_{\phi }^M{\int }_{{\mathbb{R}}^N}{|f(y)|}^2{\gamma }_k(dy).$$

(87)

Involving (83), (87), and the Riesz–Thorin interpolation theorem, we derive the following:

Proposition 4. 

For $f\in L_k^2({\mathbb{R}}^N)$ and $2\le p\le \infty$, we have

$$\parallel {\mathfrak{W}}_{\phi }^M{(f)\parallel }_{L_{{\mu }_k}^p}\le {\left({|b|}^{2⟨k⟩+N}{C}_{\phi }^M\right)}^{\frac{1}{p}}{\left({\parallel \phi \parallel }_{L_k^2}\right)}^{\frac{p-2}{p}}{\parallel f\parallel }_{L_k^2({\mathbb{R}}^N)}.$$

(88)

Theorem 3 

(Orthogonality Property). Let φ be an ADLCW. Then for all $f_1,f_2\in L_k^2({\mathbb{R}}^N)$ we have the following identity

$${|b|}^{-2⟨k⟩-N}{\int }_{{\mathbb{R}}_{+}^{N+1}}{\mathfrak{W}}_{\phi }^M(f_1)(t,r)\overline{{\mathfrak{W}}_{\phi }^M(f_2)(t,r)}d{\mu }_k(t,r)=C_{\phi }^M{\int }_{{\mathbb{R}}^N}{f}_1(x)\overline{f_2(x)}{\gamma }_k(dx).$$

(89)

We now provide a weak inversion formula for the Dunkl linear canonical wavelet transform.

Theorem 4 

($L^2$-Inversion formula). Let φ be an ADLCW. Then for every $f\in L_k^2({\mathbb{R}}^N)$, we have the weak inversion formula

$$f(y)=\frac{1}{C_{\phi }^M{|b|}^{2⟨k⟩+N}}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\mathfrak{W}}_{\phi }^M(f)(t,r)e^{\frac{-ia}{b}{|y|}^2}{\phi }_{r,t}^M(-y)d{\mu }_k(t,r)\mathrm{in}{L}_k^2({\mathbb{R}}^N),$$

which is equivalent to, for any $g\in L_k^2({\mathbb{R}}^N)$,

$$\begin{array}{c}{⟨f,g⟩}_{L_k^2}={\left(C_{\phi }^M{|b|}^{2⟨k⟩+N}\right)}^{-1}{\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\mathfrak{W}}_{\phi }^M(f)(t,r)e^{\frac{-ia}{b}{|y|}^2}{\phi }_{r,t}^M(-y)d{\mu }_k(t,r)\right)\overline{g(y)}{\gamma }_k(dy).\hfill \end{array}$$

3.1. Composition of Wavelets

Pathak [46] was the first to study the composition of wavelet transforms, and subsequently, Prasad and Kumar [47] investigated the same topic but focused on the fractional Fourier transform. In the following, we will study the composition of the DLCWT. Indeed, if $h_1$ and $h_2\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ are two ADLCWs and ${\mathfrak{W}}_{h_1}^M(f)(t,x)$ and ${\mathfrak{W}}_{h_2}^M(f)(s,y)$ are the DLCWTs of $f\in L_k^2({\mathbb{R}}^N)$, then by (85), their composition gives

$$\begin{array}{ccc}\hfill & & {\Theta }_{h_1,h_2}^M(f)(t,x,s)={\mathfrak{W}}_{h_1}^M\left({\mathfrak{W}}_{h_2}^M(f)(.,s)\right)(t,x)\hfill \\ \hfill & =& {(ibt)}^{⟨k⟩+N/2}{\int }_{{\mathbb{R}}^N}{e}^{\frac{-id}{2b}{|\xi |}^2}\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\mathfrak{F}}_D^M\left({\mathfrak{W}}_{h_2}^M(f)(.,s)\right)(\xi )({\mathfrak{F}}_D^M(h_1))(t\xi ){\gamma }_k(d\xi )\hfill \\ \hfill & =& {(ibt)}^{⟨k⟩+N/2}{(ibs)}^{⟨k⟩+N/2}{\int }_{{\mathbb{R}}^N}{e}^{\frac{-id}{b}{|\xi |}^2}\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\mathfrak{F}}_D^M(f)(\xi ){\mathfrak{F}}_D^M(h_1))(t\xi ){\mathfrak{F}}_D^M(h_2)(s\xi ){\gamma }_k(d\xi ).\hfill \end{array}$$

Thus, we can write

$$\begin{array}{ccc}\hfill ({\Theta }_{h_1,h_2}^{M}f)(t,x,s)& =& {(ts)}^{⟨k⟩+N/2}\left(f\underset{M^{-1}}{\ast }{h}_{1,t}^M\underset{M^{-1}}{\ast }{h}_{2,s}^M\right)(x),\hfill \end{array}$$

(90)

where $h_{j,z}^M$, $j=1,2$, is defined by

$${\mathfrak{F}}_D^M(h_{j,z}^M)(\xi )=e^{\frac{i}{2}\frac{d}{b}(1-z^2){|\xi |}^2}{\mathfrak{F}}_D^M(h_j)(\xi z).$$

Thus, the following relation serves as an admissibility condition for ${\Theta }_{h_1,h_2}^M$,

$$C_{h_1,h_2}^M={\int }_0^{\infty }{\int }_0^{\infty }|{\mathfrak{F}}_D^M(h_1)(t\xi ){|}^2{|{\mathfrak{F}}_D^M(h_2)(s\xi )|}^2\frac{dt}{t}\frac{ds}{s}<\infty ,$$

(91)

for $h_1,h_2\in L_k^2({\mathbb{R}}^N)$.

Theorem 5 

(Parseval-type formula). Let $h_1,h_2\in L_k^2({\mathbb{R}}^N)$ be two ADLCWs. Then, given $f,g\in L_k^2({\mathbb{R}}^N)$, we have

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }{\int }_0^{\infty }{\Theta }_{h_1,h_2}^{M}f(t,x,s)\overline{{\Theta }_{h_1,h_2}^{M}g(t,x,s)}{\gamma }_k(dx)\frac{dt}{t^{2⟨k⟩+N+1}}\frac{ds}{s^{2⟨k⟩+N+1}}\hfill \\ & =& {|b|}^{2(2⟨k⟩+N)}{C}_{h_1,h_2}^M{\int }_{{\mathbb{R}}^N}f(t)\overline{g(t)}{\gamma }_k(dt),\hfill \end{array}$$

where $C_{h_1,h_2}^M$ is defined as (91).

Proof. 

First, we assume that $h_1,h_2\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$. Using (90), the Parseval-type formula for the generalized linear canonical Fourier transform, and (72), we get

$$\begin{array}{ccc}& & {|b|}^{-2(2⟨k⟩+N)}{\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }{\int }_0^{\infty }{\Theta }_{h_1,h_2}^{M}f(t,x,s)\overline{{\Theta }_{h_1,h_2}^{M}g(t,x,s)}{\gamma }_k(dx)\frac{dt}{t^{2⟨k⟩+N+1}}\frac{ds}{s^{2⟨k⟩+N+1}}\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }{\int }_0^{\infty }{\mathfrak{F}}_D^M(f)(\xi ){\mathfrak{F}}_D^M(h_1)(t\xi ){\mathfrak{F}}_D^M(h_2)(s\xi )\overline{{\mathfrak{F}}_D^M(g)(\xi )}\overline{{\mathfrak{F}}_D^M(h_1)(t\xi )}\overline{{\mathfrak{F}}_D^M(h_2)(s\xi )}{\gamma }_k(d\xi )\frac{dt}{t}\frac{ds}{s}.\hfill \end{array}$$

From (91),

$${|b|}^{2(2⟨k⟩+N)}{C}_{h_1,h_2}^M{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi )\overline{{\mathfrak{F}}_D^M(g)(\xi )}{\gamma }_k(d\xi )={|b|}^{2(2⟨k⟩+N)}{C}_{h_1,h_2}^M{\int }_{{\mathbb{R}}^N}f(x)\overline{g(x)}{\gamma }_k(dx).$$

(92)

This completes the proof when $h_1,h_2\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$. Thus, since $L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ is dense in $L_k^2({\mathbb{R}}^N)$, then we get the result.  □

Remark 1. 

If we take $f=g$ in the last theorem, we obtain a Plancherel-type formula,

$${\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }{\int }_0^{\infty }{\left|{\Theta }_{h_1,h_2}^{M}f(t,x,s)\right|}^2{\gamma }_k(dx)\frac{dt}{t^{2⟨k⟩+N+1}}\frac{ds}{s^{2⟨k⟩+N+1}}={|b|}^{2(2⟨k⟩+N)}{C}_{h_1,h_2}^M{\parallel f\parallel }_{L_k^2}^2.$$

(93)

3.2. Time-Invariant Filter

In this subsection, we assume that $W={\mathbb{Z}}_2^N$. For any function f and $t\in {\mathbb{R}}^N$, the linear operator ${\mathfrak{L}}^M$ is said to be a time-invariant filter if it satisfies

$${\mathfrak{L}}^M(T_t^{M^{-1}}f)(s)=T_t^{M^{-1}}({\mathfrak{L}}^{M}f)(s),\mathrm{for}\mathrm{all}s\in {\mathbb{R}}^N.$$

In the following theorem, we show that the generalized convolution operator is a time-invariant filter. To prove this, we start by proving the following.

Lemma 1. 

There exists a function $g\in L_k^1({\mathbb{R}}^N)$ such that

$${\mathfrak{L}}^M({\mathfrak{B}}_k^M(\xi ,t))=\overline{{\mathfrak{F}}_D^M(g)(\xi )}{\mathfrak{B}}_k^M(\xi ,t).$$

(94)

Proof. 

Set

$$H^{\xi }(t)={\mathfrak{L}}^M\left(e^{-i\frac{a}{2b}{|\xi |}^2}{\mathfrak{B}}_k^M(\xi ,t)\right).$$

(95)

Applying operator ${\mathfrak{L}}^M$ to (61) and using (67) and (95), we get

$$\begin{array}{ccc}\hfill {\mathfrak{L}}^M\left(e^{-i\frac{a}{2b}{|\xi |}^2}{\mathfrak{B}}_k^M(\xi ,t){\mathfrak{B}}_k^M(\xi ,z)\right)& =& e^{i\frac{a}{2b}{|\xi |}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{L}}^M(e^{-i\frac{a}{2b}{|\xi |}^2}{\mathfrak{B}}_k^M(\xi ,s))e^{-i\frac{d}{2b}{(-|s|}^2-{|t|}^2-{|z|}^2)}d{\zeta }_{t,z}^k(s)\hfill \\ & =& e^{i\frac{a}{2b}{|\xi |}^2}{\int }_{{\mathbb{R}}^N}{H}^{\xi }(s)e^{-i\frac{d}{2b}{(-|s|}^2-{|t|}^2-{|z|}^2)}d{\zeta }_{t,z}^k(s).\hfill \end{array}$$

(96)

As, ${\mathfrak{L}}^M$ is linear, then by (95), we get

$${\mathfrak{L}}^M\left(e^{-i\frac{a}{2b}{|\xi |}^2}{\mathfrak{B}}_k^M(\xi ,t){\mathfrak{B}}_k^M(\xi ,z)\right)={\mathfrak{B}}_k^M(\xi ,z)H^{\xi }(t).$$

(97)

Thus, (96) and (97) give us

$$e^{i\frac{a}{2b}{|\xi |}^2}{\int }_{{\mathbb{R}}^N}{H}^{\xi }(s)e^{-i\frac{d}{2b}{(-|s|}^2-{|t|}^2-{|z|}^2)}d{\zeta }_{t,z}^k(s)={\mathfrak{B}}_k^M(\xi ,z)H^{\xi }(t).$$

(98)

Puting $t=0$ in (98) and using (30), we obtain

$$H^{\xi }(z)=e^{-i\frac{a}{2b}{|\xi |}^2}{H}^{\xi }(0){\mathfrak{B}}_k^M(\xi ,z).$$

By replacing z by t and assuming that $H^{\xi }(0)=\overline{{\mathfrak{F}}_D^M(g)(\xi )}$ for $g\in L_k^1({\mathbb{R}}^N)$, we obtain

$$H^{\xi }(t)=e^{-i\frac{a}{2b}{|\xi |}^2}\overline{{\mathfrak{F}}_D^M(g)(\xi )}{\mathfrak{B}}_k^M(\xi ,t).$$

(99)

Thus, the result follows by (95) and (99).  □

Theorem 6. 

For any function f,

$${\mathfrak{L}}^{M}f(t)=f_{a,b}\underset{M^{-1}}{\ast }g(t),$$

where g is the function given by (94) and $f_{a,b}$ the function defined by

$${\mathfrak{F}}_D^M(f_{a,b})(\xi )={(ib)}^{-⟨k⟩+N/2}{e}^{\frac{i}{2}\frac{d}{b}{|\xi |}^2}{\mathfrak{F}}_D^M(f)(\xi ).$$

Proof. 

We have

$$f(t)={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,t)}{\gamma }_k(d\xi ).$$

Then,

$${\mathfrak{L}}^{M}f(t)={\mathfrak{L}}^M\left[{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,t)}{\gamma }_k(d\xi )\right].$$

Using (94), we obtain

$$\begin{array}{ccc}\hfill {\mathfrak{L}}^{M}f(t)& =& {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi ){\mathfrak{F}}_D^M(g)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,t)}{\gamma }_k(d\xi )\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(f_{a,b}\underset{M^{-1}}{\ast }g\right)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,t)}{\gamma }_k(d\xi ).\hfill \end{array}$$

By the inversion formula, we obtain the result.  □

Theorem 7. 

Let $f\in L_k^2({\mathbb{R}}^N)$ and $h\in L_k^2({\mathbb{R}}^N)$ be an ADLCW. Then, there exists ${\mathfrak{L}}_t^M$ such that

$${\mathfrak{W}}_h^M(f)(t,x)={\mathfrak{L}}_t^{M}f(x),(t,x)\in {\mathbb{R}}_{+}^{N+1}.$$

(100)

Proof. 

By (85),

$$\begin{array}{ccc}\hfill {\mathfrak{W}}_h^M(f)(t,x)& =& {(ibt)}^{⟨k⟩+N/2}{\int }_{{\mathbb{R}}^N}{e}^{\frac{-id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M(f)(\xi ){\mathfrak{F}}_D^M(h_t^M)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi ).\hfill \end{array}$$

Then there exists a linear time-invariant filter ${\mathfrak{L}}_t^M$ such that

$${\mathfrak{L}}_t^M[\overline{{\mathfrak{B}}_k^M(\xi ,x)}]={(ibt)}^{⟨k⟩+N/2}{\mathfrak{F}}_D^M(h_t^M)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}.$$

Thus,

$${\mathfrak{W}}_h^M(f)(t,x)={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi ){\mathfrak{L}}_t^M[\overline{{\mathfrak{B}}_k^M(\xi ,x)}]{\gamma }_k(d\xi ).$$

By linearity property of ${\mathfrak{L}}_t^M$, we have

$${\mathfrak{W}}_h^M(f)(t,x)={\mathfrak{L}}_t^M\left[{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(f)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\right].$$

Hence, from the inversion formula for the linear canonical Dunkl transform, we infer

$${\mathfrak{W}}_h^M(f)(t,x)={\mathfrak{L}}_t^{M}f(x).$$

This completes the proof.  □

Theorem 8. 

Define ${\mathfrak{L}}^M$ by

$${\mathfrak{L}}^{M}f(t)=h\underset{M^{-1}}{\ast }f(t),$$

(101)

where h is a function with finite support. Then the operator ${\mathfrak{L}}^M$ is a time-invariant filter.

Proof. 

For every $t\in {\mathbb{R}}^N$,

$$\begin{array}{ccc}\hfill T_t^{M^{-1}}({\mathfrak{L}}^{M}f)(s)& =& {\int }_{{\mathbb{R}}^N}{\mathfrak{L}}^{M}f(z){\mathcal{I}}_k^M(t,s,z){\gamma }_k(dz)\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{e}^{-i\frac{d}{b}{|x|}^2}h(x)T_z^{M^{-1}}f(-x)h(z){\mathcal{I}}_{\xi }^M(t,s,z){\gamma }_k(dx){\gamma }_k(dz)\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{e}^{-i\frac{d}{b}{|x|}^2}h(x)\left({\int }_{{\mathbb{R}}^N}{T}_{-x}^{M^{-1}}f(z){\mathcal{I}}_k^M(t,s,z){\gamma }_k(dz)\right){\gamma }_k(dx)\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{e}^{-i\frac{d}{b}{|x|}^2}h(x)T_t^{M^{-1}}{T}_{-x}^{M^{-1}}f(s){\gamma }_k(dx)\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{e}^{-i\frac{d}{b}{|x|}^2}h(x)T_s^{M^{-1}}(T_t^{M^{-1}}f)(-x){\gamma }_k(dx)\hfill \\ & =& T_t^{M^{-1}}f\underset{M^{-1}}{\ast }h(s)\hfill \\ & =& {\mathfrak{L}}^M(T_t^{M^{-1}}f)(s).\hfill \end{array}$$

This completes the proof.  □

3.3. The Generalized Linear Canonical Hausdorff Operator

For $\varphi \in L^2({\mathbb{R}}_{+})$, we define the generalized Hausdorff operator $H_{\varphi }^M$ associated with the LCDT by

$$H_{\varphi }^{M}f(x):={\int }_0^{\infty }{f}_t^M(x)\varphi (t)dt,f\in L_k^2({\mathbb{R}}^N).$$

(102)

Theorem 9. 

Let $\varphi \in L^1({\mathbb{R}}_{+})$. Then for $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$, we have

$${\mathfrak{F}}_D^M(H_{\varphi }^{M}f)(\xi )={\int }_{{\mathbb{R}}_{+}}{e}^{\frac{i}{2}\frac{d}{b}(1-t^2){|\xi |}^2}{\mathfrak{F}}_D^M(f)(t\xi )\varphi (t)dt,\xi \in {\mathbb{R}}^N.$$

Proof. 

Let $\varphi \in L^1({\mathbb{R}}_{+})$, and let $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$. Then for all $\xi \in {\mathbb{R}}^N$,

$${\mathfrak{F}}_D^M(f_t^M)(\xi )=e^{\frac{i}{2}\frac{d}{b}(1-t^2){|\xi |}^2}{\mathfrak{F}}_D^M(f)(t\xi )=e^{\frac{i}{2}\frac{d}{b}(1-t^2){|\xi |}^2}{\int }_{{\mathbb{R}}^N}f(x){\mathfrak{B}}_k^M(t\xi ,x){\gamma }_k(dx).$$

(103)

Then by (102), (103) and Fubini’s theorem we obtain

$$\begin{array}{cc}\hfill {\mathfrak{F}}_D^M(H_{\varphi }^{M}f)(\xi )& ={\int }_{{\mathbb{R}}^N}{H}_{\varphi }^{M}f(x){\mathfrak{B}}_k^M(\xi ,x){\gamma }_k(dx)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}_{+}}\left[{\int }_{{\mathbb{R}}^N}{f}_t^M(x){\mathfrak{B}}_k^M(\xi ,x){\gamma }_k(dx)\right]\varphi (t)dt\hfill \\ & ={\int }_{{\mathbb{R}}_{+}}{e}^{\frac{i}{2}\frac{d}{b}(1-t^2){|\xi |}^2}{\mathfrak{F}}_D^M(f)(t\xi )\varphi (t)dt.\hfill \end{array}$$

As desired.  □

Theorem 10. 

Let ϕ be a measurable function on ${\mathbb{R}}_{+}$ such that

$$C_{\varphi }:={\int }_0^{\infty }|\varphi (r)|\frac{1}{r^{\frac{2⟨k⟩+N}{2}}}dr<\infty .$$

(104)

Then the Hausdorff operator $H_{\varphi }^M$ is bounded on $L_k^2({\mathbb{R}}^N)$, with

$$\parallel H_{\varphi }^M{f\parallel }_{L_k^2}\le C_{\varphi }{\parallel f\parallel }_{L_k^2}.$$

(105)

Proof. 

Let us note by $\eta$ the measure defined on ${\mathbb{R}}_{+}$ by

$$d\eta (r):=|\varphi (r)|dr.$$

Let us consider the integral

$${∥{\int }_0^{\infty }|f_r^M(.)|d\eta (r)∥}_{L_k^2}={\left[{\int }_0^{\infty }{\left[{\int }_{{\mathbb{R}}^N}|f_r^M(x)|d\eta (r)\right]}^2{\gamma }_k(dx)\right]}^{1/2}.$$

By Minkowski’s relation for the measure $\eta$,

$${∥{\int }_0^{\infty }|f_r^M(.)|d\eta (r)∥}_{L_k^2}\le {\int }_0^{\infty }{\parallel f_r^M(.)\parallel }_{L_k^2}d\eta (r)={\int }_{{\mathbb{R}}^N}{\left[{\int }_0^{\infty }{|f_r^M(x)|}^2{\gamma }_k(dx)\right]}^{1/2}d\eta (r).$$

Then, we obtain

$${∥{\int }_0^{\infty }|f_r^M(.)|d\eta (r)∥}_{L_k^2}\le \left[{\int }_0^{\infty }\frac{1}{r^{\frac{2⟨k⟩+N}{2}}}d\eta (r)\right]{\parallel f\parallel }_{L_k^2}=C_{\varphi }{\parallel f\parallel }_{L_k^2},$$

where

$$C_{\varphi }={\int }_0^{\infty }\frac{1}{r^{\frac{2⟨k⟩+N}{2}}}d\eta (r).$$

Moreover, by simple calculations we prove that the integral

$$H_{\varphi }^{M}f(x)={\int }_0^{\infty }{f}_r^M(x)\varphi (r)dr$$

is absolutely convergent for almost all $x\in {\mathbb{R}}^N$ and defines a function $H_{\varphi }^{M}f\in L_k^2({\mathbb{R}}^N)$ with

$$\parallel H_{\varphi }^M{f\parallel }_{L_k^2}\le {∥{\int }_0^{\infty }|f_r^M(.)|d\eta (r)∥}_{L_k^2}\le C_{\varphi }{\parallel f\parallel }_{L_k^2}.$$

The theorem is proved. □

Let $f,g\in L_k^2({\mathbb{R}}^N)$, and let $\varphi$ be a measurable function on ${\mathbb{R}}_{+}$ satisfying the condition in (104). We define the adjoint operator ${(H_{\varphi }^M)}^{\ast }$ by the relation

$${\int }_{{\mathbb{R}}^N}{(H_{\varphi }^M)}^{\ast }f(x)g(x){\gamma }_k(dx)={\int }_{{\mathbb{R}}^N}f(x)H_{\varphi }^{M}g(x){\gamma }_k(dx).$$

From Theorem 10, the operator ${(H_{\varphi }^M)}^{\ast }$ is bounded on $L_k^2({\mathbb{R}}^N)$, with

$$\parallel {(H_{\varphi }^M)}^{\ast }{f\parallel }_{L_k^2}\le C_{\varphi }{\parallel f\parallel }_{L_k^2}.$$

(106)

Theorem 11. 

Let $f\in L_k^2({\mathbb{R}}^N)$ and $\varphi \in L^1({\mathbb{R}}_{+})$, satisfying the condition in (104). Then

$${(H_{\varphi }^M)}^{\ast }(f)(x)={\int }_0^{\infty }{f}_{\frac{1}{r}}^M(x)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr.$$

Proof. 

Let $f,g\in L_k^2({\mathbb{R}}^N)$ and $\varphi \in L^1({\mathbb{R}}_{+})$, satisfying the condition in (104). Involving (102) and Fubini’s theorem, we get

$$\begin{array}{ccc}{\int }_{{\mathbb{R}}^N}f(x)H_{\varphi }^{M}g(x){\gamma }_k(dx)\hfill & =\hfill & {\int }_{{\mathbb{R}}^N}f(x)\left({\int }_0^{\infty }{g}_r^M(x)\varphi (r)dr\right){\gamma }_k(dx)\hfill \\ & =\hfill & {\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^N}{g}_r^M(x)f(x){\gamma }_k(dx)\right)\varphi (r)dr\hfill \\ & =\hfill & {\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^N}g(x)f_{\frac{1}{r}}^M(x){\gamma }_k(dx)\right)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr\hfill \\ & =\hfill & {\int }_{{\mathbb{R}}^N}g(x)\left({\int }_0^{\infty }{f}_{\frac{1}{r}}^M(x)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr\right){\gamma }_k(dx)\hfill \\ & =\hfill & {\int }_{{\mathbb{R}}^N}g(x){(H_{\varphi }^M)}^{\ast }f(x){\gamma }_k(dx),\hfill \end{array}$$

where

$${(H_{\varphi }^M)}^{\ast }f(x)={\int }_0^{\infty }{f}_{\frac{1}{r}}^M(x)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr.$$

This completes the proof of the theorem.  □

Theorem 12. 

Let $h\in L_k^2({\mathbb{R}}^N)$ be a generalized linear canonical wavelet, and let $\varphi \in L^1({\mathbb{R}}_{+})$ satisfying the condition in (104). Then for $f\in L^1\cap L_k^2({\mathbb{R}}^N)$ we have

$${\mathfrak{W}}_h^M(H_{\varphi }^{M}f)(r,s)={\int }_0^{\infty }{\mathfrak{W}}_h^M(f_t^M)(r,s)\varphi (t)dt.$$

Proof. 

Let $h\in L_k^2({\mathbb{R}}^N)$ be a generalized linear canonical wavelet, and let $f\in L^1\cap L_k^2({\mathbb{R}}^N)$. From (104) and (105), we have $H_{\varphi }^{M}f\in L_k^2({\mathbb{R}}^N)$. Then by (85) and Theorem 9, we get

$$\begin{array}{cc}\hfill {\mathfrak{W}}_h^M(H_{\varphi }^{M}f)(r,s)& ={\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M(H_{\varphi }^{M}f)(\xi ){\mathfrak{F}}_D^M(h_r^M)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,s)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{2b}{|\xi |}^2}\left[{\int }_0^{\infty }{\mathfrak{F}}_D^M(f_t^M)(\xi )\varphi (t)dt\right]{\mathfrak{F}}_D^M(h_r^M)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,s)}{\gamma }_k(d\xi )\hfill \\ & ={\int }_0^{\infty }\left[{\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M(f_t^M)(\xi ){\mathfrak{F}}_D^M(h_r^M)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,s)}{\gamma }_k(d\xi )\right]\varphi (t)dt\hfill \\ \hfill & ={\int }_0^{\infty }{\mathfrak{W}}_h^M(f_t^M)(r,s)\varphi (t)dt.\hfill \end{array}$$

This calculation is justified by the fact that

$${\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M(f_t^M)(\xi )||{\mathfrak{F}}_D^M(h_r^M)(\xi )|{\gamma }_k(d\xi )|\varphi (t)|dt\le C_{\varphi }{\parallel f\parallel }_{L_k^2}{\parallel h_r\parallel }_{L_k^2}<\infty .$$

This ends the proof of the theorem.  □

We close this subsection by giving a relation between the generalized linear canonical wavelet transform and the adjoint of the generalized Hausdorff operator.

Theorem 13. 

Let $h\in L_k^2({\mathbb{R}}^N)$ be a generalized linear canonical wavelet, and let $\varphi \in L^1({\mathbb{R}}_{+})$, satisfying the condition in (104). Then for $f\in L^1\cap L_k^2({\mathbb{R}}^N)$ we have

$${\mathfrak{W}}_h^M({(H_{\varphi }^M)}^{\ast }f)(r,s)={\int }_0^{\infty }{\mathfrak{W}}_h^M(f_{\frac{1}{r}}^M)(r,s)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr.$$

Proof. 

Let $h\in L_k^2({\mathbb{R}}^N)$ be a generalized linear canonical wavelet, and let $f,g\in L_k^2({\mathbb{R}}^N)$. From (106) we have ${(H_{\varphi }^M)}^{\ast }f\in L_k^2({\mathbb{R}}^N)$. Then by (85), Theorem 11 and Fubini’s theorem, we get

$$\begin{array}{cc}\hfill {\mathfrak{W}}_h^M({(H_{\varphi }^M)}^{\ast }f)(r,s)& ={\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{|x|}^2}{(H_{\varphi }^M)}^{\ast }f(x){\phi }_{s,r}^M(-x){\gamma }_k(dx)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{|x|}^2}\left[{\int }_0^{\infty }{f}_{\frac{1}{r}}^M(x)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr\right]{\phi }_{s,r}^M(-x){\gamma }_k(dx)\hfill \\ & ={\int }_0^{\infty }\left[{\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{|x|}^2}{f}_{\frac{1}{r}}^M(x){\phi }_{s,r}^M(-x){\gamma }_k(dx)\right]\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr\hfill \\ \hfill & ={\int }_0^{\infty }{\mathfrak{W}}_h^M(f_{\frac{1}{r}}^M)(r,s)\frac{\varphi (r)}{r^{2⟨k⟩+N}}dr.\hfill \end{array}$$

This ends the proof of the theorem.  □

4. Dunkl Linear Canonical P-Wavelet Packets

In this section, we introduce the concept of the wavelet packets and establish most of their harmonic analysis properties. From now, $g\in L_k^2({\mathbb{R}}^N)$ will be an ADLCW and ${\{{\varrho }_j\}}_{j\in \mathbb{Z}}$ a scale sequence in ${\mathbb{R}}_{+}$, strictly decreasing such that

$$\underset{j\to -\infty }{lim}{\varrho }_j=\infty \mathrm{and}\underset{j\to \infty }{lim}{\varrho }_j=0.$$

Proposition 5. 

For all $j\in \mathbb{Z}$,

(i) 

The function $\xi \mapsto {\left(\frac{1}{C_g^M}{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{|{\mathfrak{F}}_D^M(g)(t\xi )|}^2\frac{dt}{t}\right)}^{1/2}$ belongs to $L_k^2({\mathbb{R}}^N)$;

(ii) 

There exists a function $g_j^p\in L_k^2({\mathbb{R}}^N)$ such that

$${\mathfrak{F}}_D^M(g_j^p)(\xi )={\left(\frac{1}{C_g^M}{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{|{\mathfrak{F}}_D^M(g)(t\xi )|}^2\frac{dt}{t}\right)}^{1/2},$$

(107)

for almost all $t\in {\mathbb{R}}^N$.

Proof. 

From Fubini’s theorem, the Plancherel-type formula (55), and identity (80), we derive that

$$\begin{array}{cc}\hfill & {(C_g^M)}^{-1}{\int }_{{\mathbb{R}}^N}\left({\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{|{\mathfrak{F}}_D^M(g)(t\xi )|}^2\frac{dt}{t}\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={(C_g^M)}^{-1}{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}\left({\int }_{{\mathbb{R}}^N}{|{\mathfrak{F}}_D^M(g)(t\xi )|}^2{\gamma }_k(d\xi )\right)\frac{dt}{t}\hfill \\ \hfill & ={(C_g^M)}^{-1}{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{∥{\mathfrak{F}}_D^M(g_t^M)∥}_{L_k^2}^2{t}^{-1}dt={(C_g^M)}^{-1}{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{∥g_t^M∥}_{L_k^2}^2{t}^{-1}dt\hfill \\ \hfill & \le {(C_g^M)}^{-1}{∥g∥}_{L_k^2}^2{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{t}^{-2}\frac{1}{t^{2⟨k⟩+N}}dt\le {(C_g^M)}^{-1}{∥g∥}_{L_k^2}^2{\int }_{{\varrho }_{j+1}}^{{\varrho }_j}{t}^{-2⟨k⟩-N-2}dt<\infty .\hfill \end{array}$$

Consequently, assertion (ii) follows directly from Theorem 1.  □

Remark 2. 

From (76) and (107), we deduce that

$$0\le {\mathfrak{F}}_D^M(g_j^p)(\xi )\le 1\mathrm{and}\sum _{j\in \mathbb{Z}}{({\mathfrak{F}}_D^M(g_j^p)(\xi ))}^2=1,$$

(108)

for every $j\in \mathbb{Z}$ and almost every $\xi \in {\mathbb{R}}^N$.

Definition 7. 

For each $j\in \mathbb{Z}$, the function $g_j^p$ is called the DLCP-wavelet packet member of step j, and the family ${\{g_j^p\}}_{j\in \mathbb{Z}}$ is called the DLCP-wavelet packet.

Definition 8. 

Let ${\{g_j^p\}}_{j\in \mathbb{Z}}$ be a DLCP-wavelet packet. The DLCP-wavelet packet transform ${\psi }_g^{p,M}$ associated with the LCDT on ${\mathbb{R}}^N$ is defined for a function $f\in L_k^2({\mathbb{R}}^N)$ by

$${\psi }_g^{p,M}(f)(j,x)={\int }_{{\mathbb{R}}^N}f(y)g_{j,x}^{p,M}(y){\gamma }_k(dy),$$

(109)

where

$$g_{j,x}^{p,M}(y):=T_x^{M^{-1}}(g_j^p)(-y),\forall j\in \mathbb{Z}.$$

(110)

Remark 3. 

Let $f\in L_k^2({\mathbb{R}}^N)$. We observe that we can write

$${\psi }_g^{p,M}(f)(j,x)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{g}_j^p(x),\forall j\in \mathbb{Z}.$$

(111)

Some properties of these functions are summarized in the following results.

Proposition 6. 

For all $j\in \mathbb{Z}$ and $x\in {\mathbb{R}}^N$, the function $g_{j,x}^{p,M}$ belongs to $L_k^2({\mathbb{R}}^N)$ and we have

$$\parallel g_{j,x}^{p,M}{\parallel }_{L_k^2}\le {\parallel g_j^p\parallel }_{L_k^2}.$$

We also have

$${\mathfrak{F}}_D^M(g_{j,x}^{p,M})(\xi )=e^{\frac{id}{2b}{|x|}^2}\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\mathfrak{F}}_D^M(g_j^p)(\xi ).$$

(112)

Proof. 

The first inequality follows from (63), while Equation (112) follows from (42) and (66).  □

Theorem 14 

(Parseval-type formula). Let $f,h\in L_k^2({\mathbb{R}}^N)$. Then we have

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,x)\overline{{\psi }_g^{p,M}(h)(j,x)}{\gamma }_k(dx)={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}f(x)\overline{h(x)}{\gamma }_k(dx).$$

Proof. 

By (56), (72), and (111) we have, for all $j\in \mathbb{Z}$,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,x)\overline{{\psi }_g^{p,M}(h)(j,x)}{\gamma }_k(dx)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M((e^{-\frac{ia}{b}{|·|}^2}f)\underset{M^{-1}}{\ast }{g}_j^p)(\xi )\overline{{\mathfrak{F}}_D^M((e^{-\frac{ia}{b}{|·|}^2}h)\underset{M^{-1}}{\ast }{g}_j^p)(\xi )}{\gamma }_k(d\xi )\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}{|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2{\gamma }_k(d\xi ).\hfill \end{array}$$

(113)

By applying (108) and using Fubini’s theorem, we obtain

$$\begin{array}{cc}\hfill & {|b|}^{-2⟨k⟩-N}{\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,x)\overline{{\psi }_g^{p,M}(h)(j,x)}{\gamma }_k(dx)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}\left({\displaystyle \sum _{j=-\infty }^{\infty }}{|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}{\gamma }_k(d\xi )\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}f(x)\overline{h(x)}{\gamma }_k(dx)\hfill \end{array}$$

ny virtue of (56) again. Thus, the proof of the theorem is complete.  □

As a consequence of Theorem 14, we obtain the following Plancherel-type formula for the DLCP-wavelet packet transform ${\psi }_g^{p,M}$.

Corollary 1 

(Plancherel-type formula). Let $f\in L_k^2({\mathbb{R}}^N)$. Then we have

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\left|{\psi }_g^{p,M}(f)(j,x)\right|}^2{\gamma }_k(dx)={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{|f(x)|}^2{\gamma }_k(dx).$$

The following theorem gives an inversion formula for the DLCP-wavelet transform ${\psi }_g^{p,M}$.

Theorem 15. 

Let ${\{g_j^p\}}_{j\in \mathbb{Z}}$ be a DLCP-wavelet packet. For all $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)\in L_k^1({\mathbb{R}}^N)$, we have

$${(ib)}^{⟨k⟩+N/2}f(x)={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,t)\overline{g_{j,t}^{p,M}(t)}{\gamma }_k(dt),$$

for a.e. $x\in {\mathbb{R}}^N$.

Proof. 

Let

$$I^M(j,x)={\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,y)\overline{g_{j,x}^{p,M}(y)}{\gamma }_k(dy).$$

We first assume that $f\in L_k^1({\mathbb{R}}^N)$ with ${\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)\in L_k^1({\mathbb{R}}^N)$. Proposition 3 and (63) imply that the functions

$$y\mapsto \left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{g}_j^p(y)\mathrm{and}y\mapsto T_x^{M^{-1}}(g_j^p)(y)$$

belong to $L_k^2({\mathbb{R}}^N)$. Then, using (72), (110), (111), and the Parseval-type formula (56), we obtain

$$\begin{array}{cc}\hfill & I^M(j,x)={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left((e^{-\frac{ia}{b}{|·|}^2}f)\underset{M^{-1}}{\ast }{g}_j^p\right)(\xi )\overline{{\mathfrak{F}}_D^M\left(T_x^{M^{-1}}(g_j^p)\right)(\xi )}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2{\mathfrak{B}}_k^M(\xi ,x){\gamma }_k(d\xi ).\hfill \end{array}$$

(114)

Consequently, by (108), we get

$$\begin{array}{cc}\hfill {\displaystyle \sum _{j=-\infty }^{\infty }}|I^M(j,x)|& \le {|b|}^{⟨k⟩+N/2}{\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )||{\mathfrak{F}}_D^M(g_j^p)(\xi ){|}^2{\gamma }_k(d\xi ).\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )|\left({\displaystyle \sum _{j=-\infty }^{\infty }}{|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )|{\gamma }_k(d\xi )<\infty .\hfill \end{array}$$

Thus, the series above is convergent. Using (108), (114), and Fubini’s theorem, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{j=-\infty }^{\infty }}{I}^M(j,x)& ={(ib)}^{⟨k⟩+N/2}{\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){\mathfrak{B}}_k^M(\xi ,x){\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}\left(e^{-\frac{ia}{b}{|x|}^2}f(x)\right)={(ib)}^{⟨k⟩+N/2}f(x).\hfill \end{array}$$

(115)

Moreover, if $f\in L_k^2({\mathbb{R}}^N)$ and

$${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N),$$

then, by (108), we obtain

$${∥{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)·{\mathfrak{F}}_D^M(g_j^p)∥}_{L_k^2}\le {∥{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)∥}_{L_k^2}={\parallel f\parallel }_{L_k^2}<\infty .$$

Then, by Theorem 1 and equation (111), the function

$$y\mapsto \left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{g}_j^p(y),$$

belongs to $L_k^2({\mathbb{R}}^N)$, and the same process as above yields the result.  □

5. Scale-Discrete Scaling Function on ${\mathbb{R}}^N$

As a first result we have the following proposition.

Proposition 7. 

If ${\{g_j^p\}}_{j\in \mathbb{Z}}$ is a DLCP-wavelet packet, then

(i) 

For every $\xi \in {\mathbb{R}}^N$ and $J\in \mathbb{Z}$,

$${\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2=\frac{1}{C_g^M}{\int }_{{\varrho }_J}^{\infty }{\left|{\mathfrak{F}}_D^M(g)(t\xi )\right|}^2\frac{dt}{t};$$

(116)

(ii) 

For every $J\in \mathbb{Z}$, there exists a unique function $G_J^p\in L_k^2({\mathbb{R}}^N)$ (called the scale-discrete scaling function), such that

$${\mathfrak{F}}_D^M(G_J^p)(\xi )={\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\right)}^{1/2}.$$

(117)

Proof. 

Assertion (i) follows directly from (107). Let us check (ii). In view of Theorem 1, it is sufficient to check that the function

$$\xi \mapsto {\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\right)}^{1/2}$$

belongs to $L_k^2$. By Fubini’s theorem, Plancherel-type formula (55), and identity (80), we derive that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2{\gamma }_k(d\xi )\hfill \\ \hfill & ={(C_g^M)}^{-1}{\int }_{{\varrho }_J}^{\infty }{∥{\mathfrak{F}}_D^M(g_t^M)∥}_{L_k^2}^2{t}^{-1}dt={(C_g^M)}^{-1}{\int }_{{\varrho }_J}^{\infty }{∥g_t^M∥}_{L_k^2}^2{t}^{-1}dt\hfill \\ \hfill & \le {(C_g^M)}^{-1}{∥g∥}_{L_k^2}^2{\int }_{{\varrho }_J}^{\infty }\frac{1}{t^{2⟨k⟩+N+2}}dt<\infty ,\hfill \end{array}$$

which concludes the proof.  □

Remark 4. 

By (116) and (117), we have

$$0\le {\mathfrak{F}}_D^M(G_J^p)(\xi )\le 1,\underset{J\to \infty }{lim}{\mathfrak{F}}_D^M(G_J^p)(\xi )=1,$$

(118)

$${\left({\mathfrak{F}}_D^M(g_J^p)(\xi )\right)}^2={\left({\mathfrak{F}}_D^M(G_{J+1}^p)(\xi )\right)}^2-{\left({\mathfrak{F}}_D^M(G_J^p)(\xi )\right)}^2,$$

(119)

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M(G_{j+1}^p)(\xi )\right)}^2-{\left({\mathfrak{F}}_D^M(G_j^p)(\xi )\right)}^2=1,$$

(120)

for all $J\in \mathbb{Z}$ and almost all $\xi \in {\mathbb{R}}^N$.

For every $x\in {\mathbb{R}}^N$, we define the function $G_{J,x}^{p,M}$ on ${\mathbb{R}}^N$ by

$$G_{J,x}^{p,M}(·)=T_x^{M^{-1}}(G_J^p)(-·),J\in \mathbb{Z}.$$

(121)

Proposition 8. 

The function $G_{J,x}^{p,M}$ is in $L_k^2({\mathbb{R}}^N)$ and we have

$$\parallel G_{J,x}^{p,M}{\parallel }_{L_k^2}\le {\parallel G_J^p\parallel }_{L_k^2}.$$

We also have

$${\mathfrak{F}}_D^M(G_{J,x}^{p,M})(\xi )=e^{\frac{id}{2b}{|\xi |}^2}\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\mathfrak{F}}_D^M(G_J^p)(\xi ).$$

(122)

Proof. 

The proof follows along the same lines as that of Proposition 6.  □

Theorem 16. 

The Plancherel and Parseval formulas are verified for ${\{G_J^p\}}_{J\in \mathbb{Z}}$.

(i) 

For all $f,h\in L_k^2({\mathbb{R}}^N)$,

$${|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}f(x)\overline{h(x)}{\gamma }_k(dx)=\underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy).$$

(ii) 

For every $f\in L_k^2({\mathbb{R}}^N)$,

$${|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2=\underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{\left|{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\right|}^2{\gamma }_k(dy).$$

Proof. 

It follows from (69) and (121) that

$${⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}={\int }_{{\mathbb{R}}^N}f(x)G_{J,y}^{p,M}(x){\gamma }_k(dx)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{G}_J^p(y).$$

(123)

Then, by (123) and Proposition 3, we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M((e^{-\frac{ia}{b}{|·|}^2}f)\underset{M^{-1}}{\ast }{G}_j^p)(\xi )\overline{{\mathfrak{F}}_D^M((e^{-\frac{ia}{b}{|·|}^2}h)\underset{M^{-1}}{\ast }{G}_j^p)(\xi )}{\gamma }_k(d\xi )\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}{|{\mathfrak{F}}_D^M(G_j^p)(\xi )|}^2{\gamma }_k(d\xi ).\hfill \end{array}$$

(124)

Combining (118) with the dominated convergence theorem, we obtain

$$\begin{array}{cc}\hfill & \underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}{\gamma }_k(d\xi ).\hfill \end{array}$$

Equation (56) gives (i). The assertion (ii) is a consequence of (i) by taking $f=h$. □

Theorem 17 

(Plancherel-type relations).

(i) 

For all $f,g\in L_k^2({\mathbb{R}}^N)$,

$$\begin{array}{cc}\hfill {|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}f(y)\overline{g(y)}{\gamma }_k(dy)& ={\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy)\hfill \\ \hfill & +{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(j,y)\overline{{\psi }_g^{p,M}(h)(j,y)}{\gamma }_k(dy).\hfill \end{array}$$

(ii) 

For all $f\in L_k^2({\mathbb{R}}^N)$,

$$\begin{array}{c}\hfill {|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2={\int }_{{\mathbb{R}}^N}{\left|{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\right|}^2{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\left|{\psi }_g^{p,M}(f)(j,y)\right|}^2{\gamma }_k(dy).\end{array}$$

Proof. 

It follows from (117) and (124) that, for all $J\in \mathbb{Z}$, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\displaystyle \sum _{j=-\infty }^{J-1}}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}{|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2{\gamma }_k(d\xi ).\hfill \end{array}$$

On the other hand, combining this with relations (113) and (14), we obtain by (108) and (56),

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{{⟨h,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}}{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(j,y)\overline{{\psi }_g^{p,M}(h)(j,y)}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)(\xi )\overline{{\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}h)(\xi )}\left({\displaystyle \sum _{j=-\infty }^{\infty }}{|{\mathfrak{F}}_D^M(g_j^p)(\xi )|}^2\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}f(x)\overline{g(x)}{\gamma }_k(dx).\hfill \end{array}$$

Assertion (ii) is a consequence of (i) by taking $f=h$.  □

The following theorem gives two reconstruction formulas:

Theorem 18. 

For $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)\in L_k^1({\mathbb{R}}^N)$.

(i) 

For almost every $x\in {\mathbb{R}}^N$,

$${(ib)}^{⟨k⟩+N/2}f(x)=\underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy).$$

(ii) 

For almost every $x\in {\mathbb{R}}^N$ and all $J\in \mathbb{Z}$,

$$\begin{array}{cc}\hfill & {(ib)}^{⟨k⟩+N/2}f(x)={\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,t)\overline{g_{j,x}^{p,M}(t)}{\gamma }_k(dt).\hfill \end{array}$$

Proof. 

In view of (121) and (123), we have

$${⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{G}_J^p(y)\overline{T_x^{M^{-1}}(G_J^p)(y)}.$$

By using (14), (122), and Theorem 3, we get

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M((e^{-\frac{ia}{b}{|·|}^2}f)\underset{M^{-1}}{\ast }{G}_J^p)(\xi )\overline{{\mathfrak{F}}_D^M(T_x^{M^{-1}}(G_J^p))(\xi )}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){|{\mathfrak{F}}_D^M(G_J^p)(\xi )|}^2\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi ).\hfill \end{array}$$

(125)

Then, using (118) and the dominated convergence theorem, we deduce that

$$\begin{array}{cc}\hfill & \underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}\underset{J\to \infty }{lim}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{e}^{-\frac{id}{b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){|{\mathfrak{F}}_D^M(G_J^p)(\xi )|}^2\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )={(ib)}^{⟨k⟩+N/2}f(x).\hfill \end{array}$$

We now prove (ii). Using relations (117) and (125), we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\right)\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi ).\hfill \end{array}$$

(126)

It follows from the inversion formula (57), (108), and (126) that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(y)}{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,y)\overline{g_{j,x}^{p,M}(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\right)\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )={(ib)}^{⟨k⟩+N/2}f(x),\hfill \end{array}$$

for almost every $x\in {\mathbb{R}}^N$.  □

Theorem 19. 

Let $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M(e^{-\frac{ia}{b}{|·|}^2}f)\in L_k^1({\mathbb{R}}^N)$. Then for every $x\in {\mathbb{R}}^N$,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,y)\overline{g_{j,x}^{p,M}(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{j+1,y}^p}⟩}_{L_k^2}\overline{G_{j+1,y}^p(x)}{\gamma }_k(dy)-{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{j,y}^p}⟩}_{L_k^2}\overline{G_{j,y}^p(x)}{\gamma }_k(dy).\hfill \end{array}$$

(127)

Proof. 

The proof follows from (114), (119), and (125)  □

6. Dunkl Linear Canonical Modified Wavelet Packets

Let ${\{G_J^p\}}_{J\in \mathbb{Z}}$ be a scale-discrete scaling function family associated with a DLCP-wavelet packet ${\{g_j^p\}}_{j\in \mathbb{Z}}$. We define the functions $g_j^m$ and ${\tilde{g}}_j^m$, for all $j\in \mathbb{Z}$, by

$$g_j^m=G_{j+1}^p-G_j^p,{\tilde{g}}_j^m=G_{j+1}^p+G_j^p.$$

(128)

The following proposition establishes key properties of the functions $g_j^m$ and ${\tilde{g}}_j^m$.

Proposition 9.

(i) 

The functions $g_j^m$ and ${\tilde{g}}_j^m$ belong to $L_k^2({\mathbb{R}}^N)$.

(ii) 

The functions ${\mathfrak{F}}_D^M(g_j^m)$ and ${\mathfrak{F}}_D^M({\tilde{g}}_j^m)$ belong to $L_k^2({\mathbb{R}}^N)\cap L_k^{\infty }({\mathbb{R}}^N)$, such that

$$\parallel {\mathfrak{F}}_D^M(g_j^m){\parallel }_{L_k^{\infty }}\le 2and{\parallel {\mathfrak{F}}_D^M({\tilde{g}}_j^m)\parallel }_{L_k^{\infty }}\le 2.$$

(129)

(iii) 

For almost every $\xi \in {\mathbb{R}}^N$, we have

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\mathfrak{F}}_D^M(g_j^m)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^m)(\xi )=1.$$

Proof. 

These results follow directly from Proposition 7, together with identities (118), (120), and (128).  □

Definition 9. 

The sequences ${\{g_j^m\}}_{j\in \mathbb{Z}}$ and ${\{{\tilde{g}}_j^m\}}_{j\in \mathbb{Z}}$ are called the Dunkl linear canonical modified wavelet packet (or DLC$\mathfrak{M}$-wavelet) and the corresponding dual-modified wavelet packet (or dual DLC$\mathfrak{M}$-wavelet packet), respectively.

Definition 10. 

Let ${\{g_j^m\}}_{j\in \mathbb{Z}}$ be a DLC$\mathfrak{M}$-wavelet packet and ${\{{\tilde{g}}_j^m\}}_{j\in \mathbb{Z}}$ the corresponding dual DLC$\mathfrak{M}$-wavelet packet. The DLC$\mathfrak{M}$-wavelet packet transform ${\psi }_g^{m,M}$ (resp., the dual DLC$\mathfrak{M}$-wavelet packet transform ${\tilde{\psi }}_g^{m,M}$ is defined for regular functions f on ${\mathbb{R}}^N$ by

$${\psi }_g^{m,M}(f)(j,x)={\int }_{{\mathbb{R}}^N}f(y)g_{j,x}^{m,M}(y){\gamma }_k(dy)forj\in \mathbb{Z},x\in {\mathbb{R}}^N,$$

$$\left(resp.{\tilde{\psi }}_g^{m,M}(f)(j,x)={\int }_{{\mathbb{R}}^N}f(y){\tilde{g}}_{j,x}^{m,M}(y){\gamma }_k(dy)\right),$$

where

$$g_{j,x}^{m,M}(y)=T_x^{M^{-1}}(g_j^m)(-y),\left(resp.{\tilde{g}}_{j,x}^{m,M}(y)=T_x^{M^{-1}}({\tilde{g}}_j^m)(-y)\right).$$

The transform ${\psi }_g^{m,M}$ (resp. ${\tilde{\psi }}_g^{m,M}$) can also be expressed in the form

$${\psi }_g^{m,M}(f)(j,x)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{g}_j^m(x),$$

(130)

$$\left(\mathrm{resp}.{\tilde{\psi }}_g^{m,M}(f)(j,x)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{\tilde{g}}_j^m(x)\right).$$

(131)

Theorem 20 

(Plancherel-type formula). For all $f\in L_k^2({\mathbb{R}}^N)$, we have

$${|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{m,M}(f)(j,y)}{\gamma }_k(dy).$$

Proof. 

Notice first that by (130), (131), Theorem 3, and Proposition 9, the functions

$$y\mapsto {\psi }_g^{m,M}(f)(j,y)\mathrm{and}y\mapsto {\tilde{\psi }}_g^{m,M}(f)(j,y)$$

belong to $L_k^2({\mathbb{R}}^N)$ and satisfy

$$\begin{array}{cc}\hfill {\mathfrak{F}}_D^M\left({\psi }_g^{m,M}(f)(j,·)\right)& ={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right){\mathfrak{F}}_D^M(g_j^m),\hfill \\ \\ \hfill \mathrm{resp}.{\mathfrak{F}}_D^M\left({\tilde{\psi }}_g^{m,M}(f)(j,·)\right)& ={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right){\mathfrak{F}}_D^M({\tilde{g}}_j^m).\hfill \end{array}$$

From this and the Parseval formula (14), it follows that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{m,M}(f)(j,y)}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2{\mathfrak{F}}_D^M(g_j^m)(\xi )\overline{{\mathfrak{F}}_D^M({\tilde{g}}_j^m)(\xi )}{\gamma }_k(d\xi ).\hfill \end{array}$$

By using the relations (128) and (119), we show that for almost all $\xi \in {\mathbb{R}}^N$

$${\mathfrak{F}}_D^M(g_j^m)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^m)(\xi )={({\mathfrak{F}}_D^M(g_j^p)(\xi ))}^2\ge 0.$$

Then, by applying Fubini–Tonelli’s theorem and (56), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{m,M}(f)(j,y)}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\mathfrak{F}}_D^M(g_j^m)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^m)(\xi )\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2{\gamma }_k(d\xi )={|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2.\hfill \end{array}$$

Theorem 20 is proved.  □

Lemma 2. 

Let $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N)$. Then, for all $j\in \mathbb{Z}$ and $x\in {\mathbb{R}}^N$,

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,t)\overline{{\tilde{g}}_{j,x}^{m,M}(t)}{\gamma }_k(dt)& ={\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{m,M}(f)(j,t)\overline{g_{j,x}^{m,M}(t)}{\gamma }_k(dt)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,t)\overline{g_{j,x}^{p,M}(t)}{\gamma }_k(dt).\hfill \end{array}$$

Proof. 

By (14), (130), (131), and (119), we have, for all $j\in \mathbb{Z}$ and $x\in {\mathbb{R}}^N$,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,t)\overline{{\tilde{g}}_{j,x}^{m,M}(t)}{\gamma }_k(dt)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{m,M}(f)(j,t)\overline{g_{j,x}^{m,M}(t)}{\gamma }_k(dt)\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){\mathfrak{F}}_D^M(g_j^m)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^m)(\xi )\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,t)\overline{g_{j,x}^{p,M}(t)}{\gamma }_k(dt).\hfill \end{array}$$

This completes the proof.  □

Theorem 21. 

For $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N)$, we have

$$\begin{array}{cc}\hfill {(ib)}^{⟨k⟩+N/2}f(x)& ={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{m,M}(f)(j,t)\overline{{\tilde{g}}_{j,x}^{m,M}(t)}{\gamma }_k(dt)\hfill \\ \hfill & ={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,t)\overline{g_{j,x}^{m,M}(t)}{\gamma }_k(dt),\hfill \end{array}$$

for a.e. $x\in {\mathbb{R}}^N$.

Proof. 

The result follows from Theorem 15.  □

Theorem 22. 

For $f\in L_k^2({\mathbb{R}}^N)\cap L_k^1({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N)$, we have the following reconstruction formulas:

$$\begin{array}{c}\hfill {(ib)}^{⟨k⟩+N/2}f(x)={\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,t}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(t)}{\gamma }_k(dt)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{m,M}(f)(j,t){\tilde{g}}_{j,x}^{m,M}(t){\gamma }_k(dt),\end{array}$$

 and

$$\begin{array}{c}\hfill {(ib)}^{⟨k⟩+N/2}f(x)={\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,t}^{p,M}}⟩}_{L_k^2}\overline{G_{J,x}^{p,M}(t)}{\gamma }_k(dt)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{m,M}(f)(j,t)g_{j,x}^{m,M}(t){\gamma }_k(dt),\end{array}$$

for all $J\in \mathbb{Z}$ and almost every $x\in {\mathbb{R}}^N$.

Proof. 

The result follows from Theorem 18 and Lemma 2.  □

7. Dunkl Linear Canonical S-Wavelet Packet

Definition 11. 

A sequence ${\{g_j^s\}}_{j\in \mathbb{Z}}\subset L_k^2({\mathbb{R}}^N)$ is called a DLCS-wavelet packet if the following conditions hold:

(i) 

For every $j\in \mathbb{Z}$, ${\mathfrak{F}}_D^M(g_j^s)$ is real-valued.

(ii) 

For almost every $\xi \in {\mathbb{R}}^N$,

$${\gamma }_1\le {\mathfrak{F}}_D^M(g_j^s)(\xi )\le {\gamma }_2,$$

(132)

where ${\gamma }_1,{\gamma }_2$ with $0<{\gamma }_1<{\gamma }_2<\infty$.

If ${\{g_j^s\}}_{j\in \mathbb{Z}}$ is a DLCS-wavelet packet, then its corresponding dual DLCS-wavelet packet ${\{{\tilde{g}}_j^s\}}_{j\in \mathbb{Z}}$ is defined by

$${\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )=\frac{{\mathfrak{F}}_D^M(g_j^s)(\xi )}{{\displaystyle \sum _{j=-\infty }^{\infty }}{({\mathfrak{F}}_D^M(g_j^s)(\xi ))}^2},\xi \in {\mathbb{R}}^N.$$

(133)

Remark 5. 

(i) 

If ${\gamma }_1={\gamma }_2$, then ${\tilde{g}}_j^s={\gamma }_1^{-1}{g}_j^s$.

(ii) 

By (133), it is easily seen that

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\mathfrak{F}}_D^M(g_j^s)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )=1,$$

(134)

$${\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^2={\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2\right)}^{-1},$$

(135)

$$\frac{{\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^2}{{\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^2}=\frac{{\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2}{{\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2},$$

(136)

$${\displaystyle \sum _{j=-\infty }^{J-1}}{\mathfrak{F}}_D^M(g_j^s)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )=\frac{{\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2}{{\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2},$$

(137)

for all $J\in \mathbb{Z}$ and almost all $\xi \in {\mathbb{R}}^N$.

(iii) 

For almost all $\xi \in {\mathbb{R}}^N$,

$${\gamma }_2^{-1}\le {\displaystyle \sum _{j=-\infty }^{\infty }}{\left({\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^2\le {\gamma }_1^{-1},$$

where ${\gamma }_1$ and ${\gamma }_2$ are given in (132).

Definition 12. 

Let ${\{g_j^s\}}_{j\in \mathbb{Z}}$ be a DLCS-wavelet packet and ${\{{\tilde{g}}_j^s\}}_{j\in \mathbb{Z}}$ the corresponding dual DLCS-wavelet packet. The DLCS-wavelet packet transform ${\psi }_g^{s,M}$ (and the dual transform ${\tilde{\psi }}_g^{s,M}$, respectively) is defined for regular functions f on ${\mathbb{R}}^N$ by

$${\psi }_g^{s,M}(f)(j,x)={\int }_{{\mathbb{R}}^N}f(y)\overline{g_{j,x}^s(y)}{\gamma }_k(dy),j\in \mathbb{Z},x\in {\mathbb{R}}^N,$$

$$\left(resp.,{\tilde{\psi }}_g^{s,M}(f)(j,x)={\int }_{{\mathbb{R}}^N}f(y)\overline{{\tilde{g}}_{j,x}^s(y)}{\gamma }_k(dy)\right).$$

Here, for all $x,y\in {\mathbb{R}}^N$, the functions $g_{j,x}^s$ and ${\tilde{g}}_{j,x}^s$ are defined by

$$g_{j,x}^{s,M}(y)=T_x^{M^{-1}}(g_j^s)(y)\mathrm{and}{\tilde{g}}_{j,x}^{s,M}(y)=T_x^{M^{-1}}({\tilde{g}}_j^s)(y).$$

The transform ${\psi }_g^{s,M}$ (respectively, ${\tilde{\psi }}_g^{s,M}$) can be reformulated as

$${\psi }_g^{s,M}(f)(j,x)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{g}_j^s(x),$$

(138)

$$\left(\mathrm{resp}.{\tilde{\psi }}_g^{s,M}(f)(j,x)=\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\underset{M^{-1}}{\ast }{\tilde{g}}_j^s(x)\right).$$

(139)

Theorem 23 

(Plancherel-type formula). For all $f\in L_k^2({\mathbb{R}}^N)$, we have

$${|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{s,M}(f)(j,y)}{\gamma }_k(dy).$$

Proof. 

Notice first that by (138), (139), Theorem 3, and Proposition 9, the functions

$$y\mapsto {\psi }_g^{s,M}(f)(j,y)\mathrm{and}y\mapsto {\tilde{\psi }}_g^{s,M}(f)(j,y),$$

belong to $L_k^2({\mathbb{R}}^N)$ for all $j\in \mathbb{Z}$ and satisfy

$$\begin{array}{cc}\hfill {\mathfrak{F}}_D^M\left({\psi }_g^{s,M}(f)(j,·)\right)& ={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right){\mathfrak{F}}_D^M(g_j^s),\hfill \\ \\ \hfill \mathrm{resp}.{\mathfrak{F}}_D^M\left({\tilde{\psi }}_g^{s,M}(f)(j,·)\right)& ={(ib)}^{⟨k⟩+N/2}{e}^{-\frac{id}{2b}{|\xi |}^2}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right){\mathfrak{F}}_D^M({\tilde{g}}_j^s).\hfill \end{array}$$

From this and the Parseval formula (14), it follows that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{s,M}(f)(j,y)}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2{\mathfrak{F}}_D^M(g_j^s)(\xi )\overline{{\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )}{\gamma }_k(d\xi ).\hfill \end{array}$$

Then, by applying (134), (56), and Fubini–Tonelli’s theorem, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{s,M}(f)(j,y)}{\gamma }_k(dy)\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\mathfrak{F}}_D^M(g_j^s)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right){\gamma }_k(d\xi )\hfill \\ \hfill & ={|b|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\right|}^2{\gamma }_k(d\xi )={|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2.\hfill \end{array}$$

Theorem 20 is proved.  □

Theorem 24. 

For $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N)$, we have

$$\begin{array}{cc}\hfill {(ib)}^{⟨k⟩+N/2}f(x)& ={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{s,M}(f)(j,t)\overline{g_{j,x}^s(t)}{\gamma }_k(dt)\hfill \\ \hfill & ={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,t)\overline{{\tilde{g}}_{j,x}^s(t)}{\gamma }_k(dt),\hfill \end{array}$$

for a.e. $x\in {\mathbb{R}}^N$.

Proof. 

By (14), (119), (138), and (139), we have, for all $j\in \mathbb{Z}$ and $x\in {\mathbb{R}}^N$,

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{g}}_{j,x}^s(y)}{\gamma }_k(dy)={\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{s,M}(f)(j,y)\overline{g_{j,x}^s(y)}{\gamma }_k(dy)\hfill \\ & =& {(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi )\frac{{({\mathfrak{F}}_D^M(g_j^s)(\xi ))}^2}{{\displaystyle \sum _{j=-\infty }^{\infty }}{({\mathfrak{F}}_D^M(g_j^s)(\xi ))}^2}\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ & =& {(ib)}^{⟨k⟩+N/2}{e}^{\frac{ia}{b}{|x|}^2}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)(\xi ){\left({\mathfrak{F}}_D^M(g_j^p)(\xi )\right)}^2\overline{{\mathfrak{B}}_k^M(\xi ,x)}{\gamma }_k(d\xi )\hfill \\ & =& {\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,y)\overline{g_{j,x}^{p,M}(y)}{\gamma }_k(dy).\hfill \end{array}$$

Now applying (133) and Fubini’s theorem, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{g}}_{j,x}^s(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={\displaystyle \sum _{j=-\infty }^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{p,M}(f)(j,y)\overline{g_{j,x}^{p,M}(y)}{\gamma }_k(dy)={(ib)}^{⟨k⟩+N/2}f(x),\hfill \end{array}$$

a.e. on ${\mathbb{R}}^N$, again by Theorem 15.  □

Remark 6. 

Let ${\{g_j^s\}}_{j\in \mathbb{Z}}$ be a DLCS-wavelet packet and ${\{{\tilde{g}}_j^s\}}_{j\in \mathbb{Z}}$ the corresponding dual DLCS-wavelet packet. Assume that

$${\int }_{{\mathbb{R}}^N}\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M(g_j^s)(\xi )\right)}^2\right)d{\mu }_{\alpha ,\beta }(\xi )<\infty ,\forall J\in \mathbb{Z}.$$

According to (132), (135), (136), and (137), this ensures that the functions

$$\xi \mapsto {\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\left({\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^2\right)}^{1/2}\mathrm{and}\xi \mapsto {\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\mathfrak{F}}_D^M(g_j^s)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^{1/2},$$

belong to $L_k^2({\mathbb{R}}^N)$ and this enables us to state the following definition.

Definition 13. 

Let ${\{g_j^s\}}_{j\in \mathbb{Z}}$ be a DLCS-wavelet packet and ${\{{\tilde{g}}_j^s\}}_{j\in \mathbb{Z}}$ the corresponding dual DLCS-wavelet packet. We define the scale-discrete scaling function ${\{G_J^s\}}_{J\in \mathbb{Z}}$ associated with ${\{g_j^s\}}_{j\in \mathbb{Z}}$ by

$${\mathfrak{F}}_D^M(G_J^s)(\xi )={\left({\displaystyle \sum _{j=-\infty }^{J-1}}{\mathfrak{F}}_D^M(g_j^s)(\xi ){\mathfrak{F}}_D^M({\tilde{g}}_j^s)(\xi )\right)}^{1/2},\xi \in {\mathbb{R}}^N.$$

Remark 7. 

From (137), we get

$$0\le {\mathfrak{F}}_D^M(G_J^s)(\xi )\le 1\mathrm{and}\underset{J\to \infty }{lim}{\mathfrak{F}}_D^M(G_J^s)(\xi )=1,$$

for every $J\in \mathbb{Z}$ and almost every $\xi \in {\mathbb{R}}^N$.

The following two theorems can be demonstrated using the same approach as applied for Theorems 16, 17, and 18.

Theorem 25. 

For all $f\in L_k^2({\mathbb{R}}^N)$ and $J\in \mathbb{Z}$, we have the following Plancherel-type formulas:

$$\begin{array}{ccc}& & {|b|}^{N+2⟨k⟩}{\parallel f\parallel }_{L_k^2}^2=\underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{\left|{⟨f,\overline{G_{J,y}^{s,M}}⟩}_{L_k^2}\right|}^2{\gamma }_k(dy)\hfill \\ & & ={\int }_{{\mathbb{R}}^N}{\left|{⟨f,\overline{G_{J,y}^{s,M}}⟩}_{L_k^2}\right|}^2{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{\psi }}_g^{s,M}(f)(j,y)}{\gamma }_k(dy),\hfill \end{array}$$

where $G_{J,y}^{s,M}(x)=T_y^{M^{-1}}(G_J^s)(-x)$.

Theorem 26. 

For $f\in L_k^1({\mathbb{R}}^N)\cap L_k^2({\mathbb{R}}^N)$ such that ${\mathfrak{F}}_D^M\left(e^{-\frac{ia}{b}{|·|}^2}f\right)\in L_k^1({\mathbb{R}}^N)$, we have the following reconstruction formulas:

(i) 

For almost all $x\in {\mathbb{R}}^N$,

$$\begin{array}{c}\hfill f(x)=\underset{J\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{⟨f,\overline{G_{J,y}^{s,M}}⟩}_{L_k^2}\overline{G_{J,y}^{s,M}(x)}{\gamma }_k(dy).\end{array}$$

(ii) 

For almost all $x\in {\mathbb{R}}^N$ and all $J\in \mathbb{Z}$,

$$\begin{array}{cc}\hfill f(x)& ={\int }_{{\mathbb{R}}^N}{⟨f,G_{J,y}^{s,M}⟩}_{L_k^2}\overline{G_{J,y}^{s,M}(x)}{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\psi }_g^{s,M}(f)(j,y)\overline{{\tilde{g}}_{j,x}^s(y)}{\gamma }_k(dy)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{⟨f,G_{J,y}^{s,M}⟩}_{L_k^2}\overline{G_{J,y}^{s,M}(x)}{\gamma }_k(dy)+{\displaystyle \sum _{j=J}^{\infty }}{\int }_{{\mathbb{R}}^N}{\tilde{\psi }}_g^{s,M}(f)(j,y)\overline{g_{j,x}^s(y)}{\gamma }_k(dy).\hfill \end{array}$$

8. Conclusions and Perspectives

In the present paper, we have accomplished two major objectives. First, we have studied some applications for Dunkl linear canonical wavelet transforms. Then we have investigated the notion of the linear canonical Dunkl wavelet packet transform and studied its associated elementary properties. More precisely, using the harmonic analysis associated with ${\Delta }_k^M$ we define and study three types of generalized wavelet packets and their corresponding wavelet transforms in the LCD frame. The main novelty of this work is that it generalizes the theory of wavelet transforms and wavelet packets for some integral transforms, such as the Dunkl, Bessel, and Weinstein transforms [16,48,49], and covers other integral transformations, such as the Dunkl fractional transform, the Bessel fractional transform, the Dunkl–Fresnel transform, the Bessel–Fresnel transform, the Bessel LC transform, the Weinstein LC transform, and the multivariable Bessel LC transform. The classical wavelet transform is an orthogonal transform, which not only has the excellence of the Fourier transform but also settles the contradictions in the spatial field and frequency field of the Fourier transform. We also recall that wavelets are families of functions constructed from translations and dilations of a single function called the mother wavelet. These properties are not satisfied for the Dunkl wavelet transform, since, in general, we do not have the explicit expression of the main tools, such as the Dunkl kernel, the Dunkl translation, and the Dunkl transformation. The same problem is found in the linear canonical Dunkl setting. Analysis in the linear canonical Dunkl framework is more difficult since the tools and foundations on which the wavelet theory is built are complex, and we currently do not have a maple library that helps us to compare the classical case with our study. In future projects in collaboration with specialists in numerical analysis we will study the numerical part of the LCD wavelet transform and LCD wavelet packets.