Dunkl Linear Canonical Wavelet Transform: Concentration Operators and Applications to Scalogram and Localized Functions

Abstract

In the present paper we study a class of Toeplitz operators called concentration operators that are self-adjoint and compact in the linear canonical Dunkl setting. We show that a finite vector space spanned by the first eigenfunctions of such operators is of a maximal phase-space concentration and has the best phase-space concentrated scalogram inside the region of interest. Then, using these eigenfunctions, we can effectively approximate functions that are essentially localized in specific regions, and corresponding error estimates are given. These research results cover in particular the classical and the Hankel settings, and have potential application values in fields such as signal processing and quantum physics, providing a new theoretical basis for relevant research.

1. Introduction and Preliminaries

Dunkl operators are a crucial tool in the study of special functions with reflection symmetries, and harmonic analysis associated with root systems. They have progressed rapidly in recent years, and have generated considerable interest in mathematical physics. To be more precise we will fix some notation about the Dunkl transform. For more details about Dunkl theory we refer the reader to Refs. [1,2,3,4].

In this article, we keep the same notations used in Ref. [5] for the introduction of Dunkl operators. The reflection $r_{\alpha }$ in the hyperplane ${\mathcal{H}}_{\alpha }\subset {\mathbb{R}}^N$ orthogonal to $\alpha \in {\mathbb{R}}^N\setminus \left\{0\right\}$ is defined by

$$r_{\alpha }\left(y\right)=y-2{\displaystyle \frac{\langle \alpha ,y\rangle }{{|\alpha |}^2}}\alpha ,$$

(1)

where $|·|={\langle ·,·\rangle }^{1/2}$ is the usual norm on ${\mathbb{R}}^N$.

A finite subset $\mathcal{R}$ of ${\mathbb{R}}^N\setminus \left\{0\right\}$ is called a root system, if $\mathcal{R}\cap \mathbb{R}\alpha =\{\pm \alpha \}$ and if for all $\alpha \in \mathcal{R}$, $r_{\alpha }\left(\mathcal{R}\right)=\mathcal{R}$. Recall that the reflections associated to the root system $\mathcal{R}$ generate a finite group $W\subset O\left(N\right)$ called the reflection group. Then a function $k:\mathcal{R}⟶[0,\infty )$ is called a multiplicity function if it is invariant to the action of the reflection group W.

For ${\displaystyle \tilde{\alpha }\in {\mathbb{R}}^N\setminus {\cup }_{\alpha \in \mathcal{R}}{\mathcal{H}}_{\alpha }}$, we will consider the positive root system ${\mathcal{R}}_{+}=\left\{\alpha \in \mathcal{R}:\langle \alpha ,\tilde{\alpha }\rangle >0\right\}$, where for all $\alpha \in {\mathcal{R}}_{+}$, we have $\langle \alpha ,\alpha \rangle =2$. Then we define the index $\langle k\rangle$ and the function ${\omega }_k$ by

$${\displaystyle \langle k\rangle =\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right),{\omega }_k\left(t\right)=\prod _{\alpha \in {\mathcal{R}}_{+}}{|\langle \alpha ,t\rangle |}^{2k\left(\alpha \right)}.}$$

(2)

Furthermore, we define the Mehta type constant

$${\displaystyle c_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{{|t|}^2}{2}}{\omega }_k\left(t\right)dt.}$$

(3)

The Dunkl operators ${\left\{T_j\right\}}_{j\in \mathbb{N}}$ are defined by

$${\displaystyle T_{j}u(·):={\partial }_{x_j}u(·)+\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right)\langle \alpha ,e_j\rangle \frac{u(·)-u\left(r_{\alpha }(·)\right)}{\langle \alpha ,·\rangle }.}$$

(4)

where ${\left\{e_j\right\}}_{j\in \mathbb{N}}$ is an orthonormal basis of ${\mathbb{R}}^N$.

The Dunkl kernel $E_k(-i·,·)$ is the unique analytic solution on ${\mathbb{R}}^N$ of the system

$$\left\{\begin{array}{ccc}{T}_{j}u(t,z)\hfill & =\hfill & z_{j}u(t,z),j=1,\dots ,d,\hfill \\ \\ u(0,z)\hfill & =\hfill & 1,\hfill \end{array}\right.$$

(5)

where $z\in {\mathbb{R}}^N$. It satisfies

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

(6)

Moreover, it has a unique holomorphic extension to ${\mathbb{C}}^N\times {\mathbb{C}}^N$. In particular

$$E_k(z,0)=1,E_k(t,z)=E_k(z,t),E_k(\lambda z,t)=E_k(z,\lambda t),$$

for every $z,t\in {\mathbb{C}}^N$ and $\lambda \in \mathbb{C}$.

For $1\le p\le \infty$, we define $L_k^p\left({\mathbb{R}}^N\right)$ as the space of measurable functions f on ${\mathbb{R}}^N$ such that

$$\begin{array}{ccc}{\parallel u\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}\hfill & =\hfill & {\left({\displaystyle {\int }_{{\mathbb{R}}^N}{|u\left(t\right)|}^p{\gamma }_k\left(dt\right)}\right)}^{1/p}<\infty ,1\le p<\infty ,\hfill \\ \\ {\parallel u\parallel }_{L_k^{\infty }\left({\mathbb{R}}^N\right)}\hfill & =\hfill & {\displaystyle \mathrm{ess}\underset{t\in {\mathbb{R}}^N}{sup}|u\left(t\right)|<\infty ,}\hfill \end{array}$$

where ${\gamma }_k\left(dt\right):={c_k}^{-1}{\omega }_k\left(t\right)dt$. In particular the Hilbert space $L_k^2\left({\mathbb{R}}^N\right)$ is equipped with the scalar product

$${\displaystyle {\langle u,v\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}:={\int }_{{\mathbb{R}}^N}u\left(t\right)\overline{v\left(t\right)}{\gamma }_k\left(dt\right).}$$

For a function $u\in L_k^1\left({\mathbb{R}}^N\right)$, the Dunkl transform is given by

$${\displaystyle {\mathfrak{F}}_D\left(u\right)\left(z\right)={\int }_{{\mathbb{R}}^N}u\left(t\right)E_k(-it,z){\gamma }_k\left(dt\right),z\in {\mathbb{R}}^N.}$$

(7)

The Dunkl translation operator [6] is defined on $L_k^2\left({\mathbb{R}}^N\right)$ by

$${\mathfrak{F}}_D\left({\tau }_{s}u\right)=E_k(is,·){\mathfrak{F}}_D\left(u\right),s\in {\mathbb{R}}^N.$$

(8)

For a matrix $M:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ in $SL(2,\mathbb{R})$, such that $b\ne 0$, we define the linear canonical Dunkl transform (LCDT) of a function $u\in L_k^1\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{F}}_D^M\left(u\right)\left(t\right)=\frac{1}{{\left(ib\right)}^{\langle k\rangle +N/2}}{\int }_{{\mathbb{R}}^N}{\mathfrak{J}}_k^M(t,z)u\left(z\right){\gamma }_k\left(dz\right),}$$

(9)

where

$${\mathfrak{J}}_k^M(t,z)=e^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}{|t|}^2+{\displaystyle \frac{a}{b}}{|z|}^2\right)}{E}_k\left(-i{\displaystyle \frac{t}{b}},z\right).$$

(10)

This transformation is an extension of the classical linear canonical transform (LCT), which was introduced independently by Collins [7] in paraxial optics and Moshinsky–Quesne [8] in quantum mechanics. The LCT is a flexible tool for investigating deep problems in signal processing, optics, and quantum physics [9,10,11,12,13,14] and so on. Over the last years, the LCT drew the attention of many researchers and has been generalized to a wide class of integral transforms [15,16,17,18,19,20,21].

It is worth noting that all the results obtained in this paper for the LCDT remain true also for the linear canonical Hankel transform (taking $N=1$ and even functions) and the LCT (taking $k=0$), which enriches the study of the LCT and provides new results that accomplish the previous study of the LCT in the mentioned references.

The generalized translation operator in the LCDT setting is given by

$${\mathrm{T}}_y^{M,k}u\left(z\right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{(|y|}^2+{|z|}^2)}{\tau }_y\left(e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}u\right)\left(z\right),y,z\in {\mathbb{R}}^N.$$

(11)

In Ref. [22], we have introduced a new wavelet-type transformation associated to the LCDT, aiming to concentrate signals in the phase-space plane.

To be more precise, if $\phi \in L_k^2\left({\mathbb{R}}^N\right)$ is an admissible Dunkl linear canonical wavelet, satisfying ${\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1$ and

$${\displaystyle 0<C_{\varphi }^M:={\int }_0^{\infty }|{\mathfrak{F}}_D^M\left(\varphi \right)\left(\lambda y\right){|}^2\frac{d\lambda }{\lambda }<\infty ,a.ey\in {\mathbb{R}}^N,}$$

(12)

then we define the family ${\phi }_{r,t}^M$ by

$$\forall y\in {\mathbb{R}}^N,{\varphi }_{r,t}^M\left(y\right)=r^{\langle k\rangle +N/2}\overline{{\mathrm{T}}_t^{M^{-1},k}\left({\varphi }_r^M\right)(-y)},t\in {\mathbb{R}}^N,r>0,$$

(13)

where

$${\varphi }_r^M\left(y\right):=r^{-2\langle k\rangle -N}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}(1-1/r^2){|y|}^2}\varphi (y/r).$$

(14)

Therefore the Dunkl linear canonical wavelet transform (DLCWT) is defined on $L_k^2\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {{\rm Y}}_{\varphi }^M\left(u\right)(r,t):={\int }_{{\mathbb{R}}^N}u\left(y\right)\overline{{\varphi }_{r,t}^M\left(y\right)}{\gamma }_k\left(dy\right),(r,t)\in \Omega =(0,\infty )\times {\mathbb{R}}^N.}$$

(15)

In particular we have the following Plancherel-type formula for the DLCWT: For all $u\in L_k^2\left({\mathbb{R}}^N\right)$,

$${∥{{\rm Y}}_{\varphi }^M\left(u\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2=C_{\varphi }^M{|b|}^{2\langle k\rangle +N}{∥u∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(16)

where ${\mu }_k$ is the weight measure given by $d{\mu }_k(r,t)=r^{-2\langle k\rangle -N-1}{\gamma }_k\left(dt\right)dr$ and for $1⩽p⩽\infty$, $L_{{\mu }_k}^p(\Omega )$ is the space of measurable functions f on $\Omega$ such that

$$\begin{array}{ccc}{\parallel f\parallel }_{L_{{\mu }_k}^p(\Omega )}\hfill & =\hfill & {\left({\displaystyle {\int }_{\Omega }{|f(r,t)|}^{p}d{\mu }_k(r,t)}\right)}^{1/p}<\infty ,1\le p<\infty ,\hfill \\ \\ {\parallel f\parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}\hfill & =\hfill & \underset{(r,t)\in \Omega }{\mathrm{ess} \sup } |f(r,t)|<\infty ,\hfill \end{array}$$

We define the phase-space localization (or Toeplitz) operator associated to the DLCWT on $L_k^2\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{T}}_{\varphi }^M\left(\sigma \right)\left(f\right)\left(x\right)=\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }\sigma (r,t){{\rm Y}}_{\varphi }^M\left(f\right)(r,t){\varphi }_{r,t}^M\left(x\right)d{\mu }_k(r,t),x\in {\mathbb{R}}^N,}$$

(17)

where $c(b,k,\varphi )=C_{\varphi }^M{|b|}^{2\langle k\rangle +N}$ and $\sigma$ is a measurable function on $\Omega$ (called symbol). This definition is frequently easier to understand in a weak sense, i.e., for $u,v\in L_k^2\left({\mathbb{R}}^N\right)$,

$${\displaystyle {〈{\mathfrak{T}}_{\varphi }^M\left(\sigma \right)\left(u\right),v〉}_{L_k^2\left({\mathbb{R}}^N\right)}=\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }\sigma \left(z\right){{\rm Y}}_{\varphi }^M\left(u\right)\left(z\right)\overline{{{\rm Y}}_{\varphi }^M\left(v\right)\left(z\right)}d{\mu }_k\left(z\right).}$$

(18)

The adjoint of linear operator ${\mathfrak{T}}_{\varphi }^M\left(\sigma \right):L_k^2\left({\mathbb{R}}^N\right)\to L_k^2\left({\mathbb{R}}^N\right)$ is ${\mathfrak{T}}_{\varphi }^M\left(\overline{\sigma }\right):L_k^2\left({\mathbb{R}}^N\right)\to L_k^2\left({\mathbb{R}}^N\right)$, that is,

$${\left({\mathfrak{T}}_{\varphi }^M\right)}^{*}\left(\sigma \right)={\mathfrak{T}}_{\varphi }^M\left(\overline{\sigma }\right).$$

(19)

Time-frequency localization operators were initially introduced and studied by Daubechies [23] and Ramanathan–Topiwala [24]. These operators can be used to find and extract a signal’s components from its time-frequency plane representation. They have been employed in the approximation of pseudo-differential operators [25] and in physics as anti-Wick operators, as tools for quantization processes [26].

In Ref. [22] we have studied the boundedness and compactness of localization operators ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ in general setting, with two admissible wavelets and in the $L^p$-spaces. In this paper we only focus on the $L^2$-space, with only one admissible wavelet and with a special symbol. In particular, we have proof that for any symbol $\sigma \in L_{{\mu }_k}^p(\Omega )$, $1⩽p⩽\infty$, ${\mathfrak{T}}_{\varphi }^M\left(\sigma \right):L_k^2\left({\mathbb{R}}^N\right)⟶L_k^2\left({\mathbb{R}}^N\right)$ is bounded, with

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\sigma \right)\parallel }_{S_{\infty }}⩽{\left(c(b,k,\varphi )\right)}^{-{\displaystyle \frac{1}{p}}}{\parallel \sigma \parallel }_{L_{{\mu }_k}^p(\Omega )}.$$

(20)

and belongs to the Schatten class $S_p$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\sigma \right)\parallel }_{S_p}⩽{\left(c(b,k,\varphi )\right)}^{-{\displaystyle \frac{1}{p}}}{\parallel \sigma \parallel }_{L_{{\mu }_k}^p(\Omega )}.$$

(21)

If we choose $\sigma ={\chi }_R$, where R is a subset of $\Omega$, with finite measure $0<{\mu }_k\left(R\right)<\infty ,$ then the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\sigma \right)$ is known as concentration operator, which will be notated as ${\mathcal{L}}_{\varphi }^M\left(R\right)$.

Let $P_{\varphi }^M$ he orthogonal projection from $L_{{\mu }_k}^2(\Omega )$ onto ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ defined by

$$P_{\varphi }^M:={{\rm Y}}_{\varphi }^M{\left({{\rm Y}}_{\varphi }^M\right)}^{*},$$

(22)

and $P_R$ the orthogonal projection from $L_{{\mu }_k}^2(\Omega )$ onto the subspace of functions supported in $R\subset {\mathbb{R}}^N$, i.e.,

$$P_{R}F={\chi }_{R}F,F\in L_{{\mu }_k}^2(\Omega ),$$

(23)

where ${\chi }_R$ is the characteristic function on R and ${\left({{\rm Y}}_{\varphi }^M\right)}^{*}$ is the adjoint of ${{\rm Y}}_{\varphi }^M$ defined on $L_{{\mu }_k}^2(\Omega )$ onto $L_{{\mu }_k}^2(\Omega )$ by

$${〈{{\rm Y}}_{\varphi }^M\left(R\right),v〉}_{L_{{\mu }_k}^2(\Omega )}=c(b,k,\varphi ){〈u,{\left({{\rm Y}}_{\varphi }^M\right)}^{*}\left(v\right)〉}_{L_k^2\left({\mathbb{R}}^N\right)},u\in L_k^2\left({\mathbb{R}}^N\right),v\in L_{{\mu }_k}^2(\Omega ).$$

(24)

We will prove that ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is a reproducing kernel Hilbert space, with kernel function

$${\displaystyle {\mathcal{K}}_{\varphi }^M(r^{\prime },x^{\prime };r,x):=\frac{1}{c(b,k,\varphi )}{\int }_{{\mathbb{R}}^N}{\varphi }_{r^{\prime },x^{\prime }}^M\left(y\right)\overline{{\varphi }_{r,x}^M\left(y\right)}{\gamma }_k\left(dy\right),}$$

(25)

and show that the Calderón–Toeplitz-type operator ${\mathfrak{C}}_{\varphi ,R,M}=P_{\varphi }^M{P}_R:{{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\to {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ and the concentration operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ are related by

$${\mathfrak{C}}_{\varphi ,R,M}={{\rm Y}}_{\varphi }^M{\mathcal{L}}_{\varphi }^M\left(R\right){\left({{\rm Y}}_{\varphi }^M\right)}^{*}.$$

(26)

Moreover we will show that ${\mathfrak{C}}_{\varphi ,R,M}$ belongs to $S_1$, such that

$${\displaystyle tr\left({\mathfrak{C}}_{\varphi ,R,M}\right)=tr\left({\mathcal{L}}_{\varphi }^M\left(R\right)\right)={\int }_R{\parallel {\varphi }_{r,x}^M\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{2}d{\mu }_k(r,x).}$$

(27)

Let ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, $\epsilon \in (0,1)$ be the set of all functions $f\in L_k^2\left({\mathbb{R}}^N\right)$ that are $(\epsilon ,\varphi )$-concentrated in R, i.e., functions $f\in L_k^2\left({\mathbb{R}}^N\right)$ satisfying

$${∥P_{R^c}{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\le c(b,k,\varphi )\epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(28)

where $R^c$ is the complement of R in ${\mathbb{R}}^N$. Therefore we obtain the following Donoho–Stark-type uncertainty principle [27] for the DLCWT, that is, if $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then R satisfies

$${\mu }_k\left(R\right)\ge c(b,k,\varphi )(1-\epsilon ),$$

(29)

which means that the support (in the case $\epsilon =0$) or the essential support (in the case $0<\epsilon <1$) of the DLCWT of a nonzero function cannot be too small. Moreover we will show that a function $f\in L_k^2\left({\mathbb{R}}^N\right)$ belongs to ${\mathcal{C}}_{\epsilon ,\phi }\left(R\right)$ if and only if

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right),f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\ge (1-\epsilon ){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(30)

We say that $f\in L_k^2\left({\mathbb{R}}^N\right)\setminus \left\{0\right\}$ is $\epsilon$-localized with respect to the operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ if

$$\parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)\left(f\right)-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\le \epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(31)

Therefore, we will prove that, if f is $\epsilon$-localized with respect to ${\mathcal{L}}_{\varphi }^M\left(R\right)$, then $f\in {\mathcal{C}}_{\epsilon +\sqrt{\epsilon },\varphi }\left(R\right)$, and conversely, if $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then f is $\epsilon$-localized with respect to ${\mathcal{L}}_{\varphi }^M\left(R\right)$. Moreover, we will show the uncertainty inequality for the concentration operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$: If $f\in L_k^2\left({\mathbb{R}}^N\right)$ is ${\epsilon }_1$-localized with respect to ${\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)$ and ${\epsilon }_2$-localized with respect to ${\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)$. Then if $\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}<1$ we have

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)\left(1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}\right),$$

(32)

Consequently, if $f\in L_k^2\left({\mathbb{R}}^N\right)$ is ${\epsilon }_1$-localized with respect to ${\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)$ and ${\epsilon }_2$-localized with respect to ${\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)$, we have

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)(1-{\epsilon }_1-\sqrt{{\epsilon }_1})(1-{\epsilon }_2-\sqrt{{\epsilon }_2}),$$

(33)

where ${\epsilon }_1+\sqrt{{\epsilon }_1}<1$ and ${\epsilon }_2+\sqrt{{\epsilon }_2}<1$. Additionally, if $f\in {\mathcal{C}}_{{\epsilon }_1,{\varphi }_1}\left(R_1\right)\cap {\mathcal{C}}_{{\epsilon }_2,{\varphi }_2}\left(R_2\right)$, then

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)(1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}),$$

(34)

where $\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}<1$.

Now since the Toeplitz operator ${\mathcal{L}}_{\varphi }^M\left(R\right)={\left({{\rm Y}}_{\varphi }^M\right)}^{*}{\chi }_R{{\rm Y}}_{\varphi }^M$ that we consider is a compact and self-adjoint operator, the spectral theorem gives the spectral representation

$${\displaystyle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right)=\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){〈f,{\psi }_j^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}{\psi }_j^R,f\in L_k^2\left({\mathbb{R}}^N\right),}$$

(35)

where ${\left\{{\alpha }_j\left(R\right)\right\}}_{j=1}^{\infty }$ are the positive eigenvalues arranged in a nonincreasing manner and ${\left\{{\psi }_j^R\right\}}_{j=1}^{\infty }$ is the corresponding orthonormal sequence of eigenfunctions. We have ${\alpha }_j\left(R\right)\searrow 0$ and from Equation (20), for $p=\infty$, the eigenvalues satisfy

$${\alpha }_j\left(R\right)\le {\alpha }_1\left(R\right)\le 1,\forall j\ge 1.$$

(36)

Notice that by Equation (35), the concentration operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ is positive and satisfies

$${〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left(u\right),u〉}_{L_k^2\left({\mathbb{R}}^N\right)}=\sum _{i=1}^{\infty }{\alpha }_j\left(R\right){\left|{〈u,{\psi }_j^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2={\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(u\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2.$$

(37)

This shows that the concentration operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ is useful in studying the concentration problem

$$\mathrm{Maximize}{∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(u\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2,{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$$

(38)

It follows that the first eigenfunction ${\psi }_1^R$ of the compact self-adjoint operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ solves the problem (38) since

$${\alpha }_1\left(R\right)={\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_R{{\rm Y}}_{\varphi }^M\left({\psi }_1^R\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2=max\left\{{\langle {\mathcal{L}}_{\varphi }^M\left(R\right)u,u\rangle }_{L_k^2\left({\mathbb{R}}^N\right)},{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1\right\}.$$

Thus, ${\psi }_1^R$ has the maximum energy inside R which is given by ${\alpha }_1\left(R\right)$. Furthermore, according to the min-max lemma, we have, for $j\ge 2$

$$\begin{array}{ccc}\hfill {\alpha }_j\left(R\right)& =& {\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_R{{\rm Y}}_{\varphi }^M\left({\psi }_j^R\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\hfill \\ & =& max\left\{{\langle {\mathcal{L}}_{\varphi }^M\left(R\right)u,u\rangle }_{L_k^2\left({\mathbb{R}}^N\right)},{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1,u\perp {\psi }_1^R,\dots ,{\psi }_{j-1}^R\right\}.\hfill \end{array}$$

Hence, the eigenvalues of ${\mathcal{L}}_{\varphi }^M\left(R\right)$ determines the number of orthogonal functions that are well concentrated in R. Moreover, if the eigenvalue ${\alpha }_j\left(R\right)$ satisfies

$${\alpha }_j\left(R\right)\ge (1-\epsilon ),$$

(39)

then from Equation (37)

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_j^R,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}={\alpha }_j\left(R\right)\ge (1-\epsilon ).$$

(40)

Therefore by Equation (30), any eigenfunction ${\psi }_j^R$ satisfying Equation (39) is in ${\mathcal{C}}_{\epsilon ,\psi }\left(R\right)$.

We define the scalogram of a function $u\in L_k^2\left({\mathbb{R}}^N\right)$ with respect to $\varphi$ by

$${\mathbf{S}}_{\varphi ,M}^D\left(u\right)={\left|{{\rm Y}}_{\varphi }^M\left(u\right)\right|}^2.$$

(41)

Then from Equation (16), we have

$${\displaystyle {\int }_{\Omega }{\mathbf{S}}_{\varphi ,M}^D\left(u\right)\left(z\right)d{\mu }_k\left(z\right)=c(b,k,\varphi ){\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

(42)

This justifies the interpretation of the scalogram as a time-frequency energy density, and by Equation (18) we have

$${\displaystyle {〈{\mathcal{L}}_{\varphi }^M\left(R\right)u,u〉}_{L_k^2\left({\mathbb{R}}^N\right)}=\frac{1}{c(b,k,\varphi )}{\int }_R{\mathbf{S}}_{\varphi ,M}^D\left(u\right)\left(z\right)d{\mu }_k\left(z\right).}$$

(43)

We define the time-frequency concentration of the subspace V in R by

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V\right):=\frac{1}{n}\sum _{i=1}^n{\int }_R{\mathbf{S}}_{\varphi ,M}^D\left(v_i\right)\left(z\right)d{\mu }_k\left(z\right),}$$

(44)

where V is a subspace of $L_k^2\left({\mathbb{R}}^N\right)$ of dimension n, and ${\left\{v_i\right\}}_{i=1}^n$ is an orthonormal basis of V. Then we will prove that n-dimensional signal space $V_n=\mathrm{span}{\left\{{\psi }_i^R\right\}}_{i=1}^n$ generated by the first n eigenfunctions of the concentration operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ corresponding to the n largest eigenvalues ${\left\{{\alpha }_i\left(R\right)\right\}}_{i=1}^n$ maximize the regional concentration ${\zeta }_{R,\varphi }^M\left(V\right)$ and we have

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V_n\right):=\frac{1}{n}\sum _{i=1}^n{\alpha }_i\left(R\right).}$$

(45)

Now for any function f belonging to $V_n$ we have

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right)f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\ge {\alpha }_n\left(R\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(46)

which means that $V_{d(\epsilon ,R)}$ is a subspace of ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, where

$$d(\epsilon ,R):=\mathrm{card}\left\{j:{\alpha }_j\left(R\right)\ge 1-\epsilon \right\},\epsilon \in (0,1).$$

(47)

On the other hand, since

$${〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left({\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_n^R\right),{\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_j^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}={\alpha }_j\left(R\right){∥{\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_j^R∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(48)

then by Equation (30), the function ${\mathcal{L}}_{\varphi }^M\left(R\right)\left({\psi }_j^R\right)$ is in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, if and only if $j\le d(\epsilon ,R)$. Thus, if $j\ge 1+d(\epsilon ,R)$, then neither ${\psi }_j^R$ nor ${\mathcal{L}}_{\varphi }^M\left(R\right)\left({\psi }_j^R\right)$ are in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$. Hence for a properly chosen of n, functions in $V_n$ have the best time-frequency concentrated scalogram inside R. Moreover the quantity $d(\epsilon ,R)$ determines the maximum dimension of a subspace $V\subset L_k^2\left({\mathbb{R}}^N\right)$ consisting of signals that have a well-concentrated scalogram in R.

In contrast, functions in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ are not necessarily in $V_{d(\epsilon ,R)}$. Nevertheless we will prove the result characterizing functions that are in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, that is, a function $f\in L_k^2\left({\mathbb{R}}^N\right)$ belongs to ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ if, and only if, it satisfies

$$\begin{array}{cc}\hfill \sum _{j=1}^{d(\epsilon ,R)}({\alpha }_j\left(R\right)-1+\epsilon ){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2& \ge (1-\epsilon ){∥{\mathcal{P}}_{\mathrm{Ker}}\left(f\right)∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ \hfill & +\sum _{j=1+d(\epsilon ,R)}^{\infty }(1-\epsilon -{\alpha }_j\left(R\right)){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2,\hfill \end{array}$$

(49)

where ${\mathcal{P}}_{\mathrm{Ker}}\left(f\right)$ denotes the orthogonal projection of f onto the kernel of ${\mathcal{L}}_{\varphi }^M\left(R\right)$.

While a function $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ does not necessarily lies in some subspace $V_n$ of eigenfunctions, it can be approximated using a finite number of such eigenfunctions, that is, we will show that, if $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then,

$${∥f-\sum _{j=1}^{d(\tilde{\epsilon },R)}{〈f,{\psi }_j^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}{\psi }_j^R∥}_{L_k^2\left({\mathbb{R}}^N\right)}\le \sqrt{{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)},$$

(50)

where $0<\tilde{\epsilon }<1$ is a fixed real number.

2. Dunkl Linear Canonical Wavelet Transform (DLCWT) and Toeplitz Operators

In this section, we will recall the basic definitions about the linear canonical Dunkl transform in order to define the Dunkl linear canonical wavelet transform and give its well-known properties. Then we will introduce the notion of Toeplitz (or localization) operators related to the DLCWT, and study their boundedness and compactness. Throughout this paper $M:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is a matrix in $SL(2,\mathbb{R})$, such that $b\ne 0$. Notice that its inverse $M^{-1}$ is given by $\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$. The LCDT of a function $u\in L_k^1\left({\mathbb{R}}^N\right)$ is given by

$${\displaystyle {\mathfrak{F}}_D^M\left(u\right)\left(t\right)=\frac{1}{{\left(ib\right)}^{\frac{2\langle k\rangle +N}{2}}}{\int }_{{\mathbb{R}}^N}{\mathfrak{J}}_k^M(t,z)u\left(z\right){\gamma }_k\left(dz\right),}$$

(51)

where

$${\mathfrak{J}}_k^M(t,z)=e^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}{|t|}^2+{\displaystyle \frac{a}{b}}{|z|}^2\right)}{E}_k\left(-i{\displaystyle \frac{t}{b}},z\right).$$

(52)

The LCDT was first defined and studied in Ref. [15], then generalized and improved in Ref. [28]. We have, for every $t,z\in {\mathbb{R}}^N$,

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

(53)

Theorem 1 

(Riemann–Lebesgue-type Inequality). For every $u\in L_k^1\left({\mathbb{R}}^N\right)$, its LCDT belongs to $C_0\left({\mathbb{R}}^N\right)$ such that

$$\parallel {\mathfrak{F}}_D^M{\left(u\right)\parallel }_{L_k^{\infty }\left({\mathbb{R}}^N\right)}\le {|b|}^{-(\langle k\rangle +N/2)}{\parallel u\parallel }_{L_k^1\left({\mathbb{R}}^N\right)}.$$

(54)

Proposition 1

(Plancherel-type Formulas).

1. 

For every $u,v\in L_k^1\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(u\right)\left(t\right)\overline{v\left(t\right)}{\gamma }_k\left(dt\right)={\int }_{{\mathbb{R}}^N}u\left(t\right)\overline{{\mathfrak{F}}_D^{M^{-1}}\left(v\right)\left(t\right)}{\gamma }_k\left(dt\right).}$$

(55)

2. 

If u belongs to $L_k^1\left({\mathbb{R}}^N\right)\cap L_k^2\left({\mathbb{R}}^N\right)$, then ${\mathfrak{F}}_D^M\left(u\right)$ belongs to $L_k^2\left({\mathbb{R}}^N\right)$ such that

$$\parallel {\mathfrak{F}}_D^M{\left(u\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(56)

3. 

The LCDT has a unique extension to an isometric isomorphism on $L_k^2\left({\mathbb{R}}^N\right)$, which still denoted by ${\mathfrak{F}}_D^M$.

4. 

For any $u,v\in L_k^2\left({\mathbb{R}}^N\right)$,

$${\langle {\mathfrak{F}}_D^M\left(v\right),u\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}={\langle v,{\mathfrak{F}}_D^{M^{-1}}u\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(57)

5. 

If $u\in L_k^1\left({\mathbb{R}}^N\right)$ with ${\mathfrak{F}}_D^M\left(u\right)\in L_k^1\left({\mathbb{R}}^N\right),$ then

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

(58)

Definition 1.

The generalized translation operator associated for the LCDT [28] is given by

$$\begin{array}{c}\hfill {\mathrm{T}}_y^{M,k}u(z)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{(|y|}^2+{|z|}^2)}{\tau }_y\left(e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}u\right)(z).\end{array}$$

(59)

It satisfies the properties for $x,y,z\in {\mathbb{R}}^N$, we have

1.

${\mathrm{T}}_0^{M,k}=\mathrm{Id}$ and ${\mathrm{T}}_y^{M,k}u\left(z\right)={\mathrm{T}}_z^{M,k}u\left(y\right).$

2.

The translation product formula is given by

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{T}}_y^{M,k}\left({\mathfrak{J}}_k^M(·,x)\right)\left(z\right)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}{|x|}^2}{\mathfrak{J}}_k^M(y,x){\mathfrak{J}}_k^M(z,x).\hfill \end{array}\end{array}$$

(60)

3.

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

$${∥{\mathrm{T}}_y^{M,k}u∥}_{L_k^2\left({\mathbb{R}}^N\right)}\le {∥u∥}_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(61)

4.

If $u\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right),$ (resp. $L_k^2\left({\mathbb{R}}^N\right)$), we have

$${\mathfrak{F}}_D^M\left({\mathrm{T}}_y^{M^{-1},k}u\right)(·)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}\overline{{\mathfrak{J}}_k^M(y,·)}{\mathfrak{F}}_D^M\left(u\right)(·).$$

(62)

Definition 2.

The generalized convolution product associated with ${\mathfrak{F}}_D^M$ of two suitable functions u and v on ${\mathbb{R}}^N$, is the function $u\underset{M}{\odot }v$ defined by

$${\displaystyle u\underset{M}{\odot }v\left(y\right)={\int }_{{\mathbb{R}}^N}{\mathrm{T}}_y^{M,k}u(-t)e^{-i\frac{d}{b}{|t|}^2}v\left(t\right){\gamma }_k\left(dt\right).}$$

(63)

Then we have the properties

1.

$u\underset{M}{\odot }v=v\underset{M}{\odot }u.$

2.

${\mathrm{T}}_y^{M,k}\left(u\underset{M}{\odot }v\right)=\left({\mathrm{T}}_y^{M,k}u\right)\underset{M}{\odot }v=u\underset{M}{\odot }\left({\mathrm{T}}_y^{M,k}v\right).$

Proposition 2.

Let $p\in (1,2]$.

1. 

For $v\in L_k^1\left({\mathbb{R}}^N\right)$ and $u\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$,

$${\left(ib\right)}^{-\langle k\rangle -N/2}{\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{\odot }v\right)(·)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}{\mathfrak{F}}_D^M\left(u\right)(·){\mathfrak{F}}_D^M\left(v\right)(·).$$

(64)

2. 

For $v\in L_k^p\left({\mathbb{R}}^N\right)$ and $u\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$,

$${\left(ib\right)}^{-\langle k\rangle -N/2}{\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{\odot }v\right)(·)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}{\mathfrak{F}}_D^M\left(u\right)(·){\mathfrak{F}}_D^M\left(v\right)(·),a.e.$$

(65)

3. 

For $u,v,w\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$, we have

$$\left(u\underset{M}{\odot }v\right)\underset{M}{\odot }w=u\underset{M}{\odot }\left(v\underset{M}{\odot }w\right).$$

(66)

4. 

It is not necessary for the functions to be radials for the preceding three results to hold if $W={\mathbb{Z}}_2^N$.

5. 

For all $u,v\in L_k^2\left({\mathbb{R}}^N\right)$, we have

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

(67)

2.1. DLCWT

In this section, we introduce the generalized linear canonical wavelet transform in the linear canonical Dunkl setting, and we give some of its properties.

Definition 3.

We say that $\varphi \in L_k^2\left({\mathbb{R}}^N\right)$ is an admissible Dunkl linear canonical wavelet if for almost $y\in {\mathbb{R}}^N$

$${\displaystyle 0<C_{\varphi }^M:={\int }_0^{\infty }|{\mathfrak{F}}_D^M\left(\varphi \right)\left(\lambda y\right){|}^2\frac{d\lambda }{\lambda }<\infty .}$$

(68)

Let $r>0$, $t\in {\mathbb{R}}^N$ and $\varphi \in L_k^2\left({\mathbb{R}}^N\right)$. We define the family ${\varphi }_{r,t}^M$ by

$${\varphi }_{r,t}^M\left(y\right)=r^{\langle k\rangle +N/2}\overline{{\mathrm{T}}_t^{M^{-1},k}\left({\varphi }_r^M\right)(-y)},y\in {\mathbb{R}}^N,$$

(69)

where

$${\varphi }_r^M\left(y\right):=r^{-2\langle k\rangle -N}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}(1-1/r^2){|y|}^2}\varphi (y/r).$$

(70)

Proposition 3.

Let $(r,t)\in \Omega$ and $\lambda \in {\mathbb{R}}^N$.

1. 

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

$${\parallel {\varphi }_{r,t}^M\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\le {\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(71)

2. 

For all $\varphi \in L_k^1\left({\mathbb{R}}^N\right)\cup L_k^2\left({\mathbb{R}}^N\right)$,

$${\mathfrak{F}}_D^M\left({\varphi }_r^M\right)\left(\lambda \right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}(1-1/r^2){|\lambda |}^2}{\mathfrak{F}}_D^M\left(\varphi \right)\left(r\lambda \right).$$

(72)

3. 

For all $\varphi \in L_k^1\left({\mathbb{R}}^N\right)\cup L_k^2\left({\mathbb{R}}^N\right)$,

$${\mathfrak{F}}_D^M\left(\overline{{\varphi }_{r,t}^M}\right)\left(\lambda \right)=r^{\langle k\rangle +N/2}{e}^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}(1-1/r^2){|\lambda |}^2-{\displaystyle \frac{a}{b}}{|t|}^2\right)}{\mathfrak{J}}_k^M(-t,\lambda /b){\mathfrak{F}}_D^M\left(\varphi \right)\left(r\lambda \right).$$

(73)

Proof. 

The first assertion is proved by using Equations (61), (69) and (70). Involving Equations (51) and (70), we infer Equation (72). Using Equation (62), Equations (69) and (72) we derive Equation (73). □

Definition 4.

The Dunkl linear canonical wavelet transform (DLCWT) of any regular function $u\in L_k^2\left({\mathbb{R}}^N\right)$ is denoted by ${{\rm Y}}_{\varphi }^M\left(u\right)$ and is defined as

$${\displaystyle {{\rm Y}}_{\varphi }^M\left(u\right)(r,t):={\int }_{{\mathbb{R}}^N}u\left(y\right)\overline{{\varphi }_{r,t}^M\left(y\right)}{\gamma }_k\left(dy\right),(r,t)\in \Omega ,}$$

(74)

where ${\varphi }_{r,t}^M$ is given by Equation (69).

The last formula can also be written as

$${{\rm Y}}_{\varphi }^M\left(u\right)(r,t)=r^{{\displaystyle \frac{2\langle k\rangle +N}{2}}}{\mathbf{L}}_{-{\displaystyle \frac{2a}{b}}}u\underset{M^{-1}}{\odot }{\varphi }_r^M\left(t\right),$$

(75)

where ${\mathbf{L}}_s$, $s\in \mathbb{R}$ is the chirp multiplication operator given by

$${\mathbf{L}}_{s}u(·)=e^{{\displaystyle \frac{is}{2}}{|·|}^2}u(·).$$

(76)

By Equation (71), Equation (74) and Cauchy–Schwartz’s inequality, we have for any $u\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\parallel {{\rm Y}}_{\varphi }^M{\left(u\right)\parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}\le {\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(77)

Moreover, if $\varphi \in L_k^1\left({\mathbb{R}}^N\right)\cap L_k^2\left({\mathbb{R}}^N\right)$ and $u\in L_k^2\left({\mathbb{R}}^N\right)$, then by Equations (65) and (75)

$${\mathfrak{F}}_D^M\left({{\rm Y}}_{\varphi }^M\left(u\right)(r,·)\right)\left(\lambda \right)={\left(ibr\right)}^{\langle k\rangle +N/2}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|\lambda |}^2}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}u\right)\left(\lambda \right){\mathfrak{F}}_D^M\left({\varphi }_r^M\right)\left(\lambda \right).$$

(78)

Theorem 2

(Plancherel-type formula). Let ϕ be an admissible Dunkl linear canonical wavelet. Then for all $u\in L_k^2\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{{\rm Y}}_{\varphi }^M{\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)=c(b,k,\varphi ){\int }_{{\mathbb{R}}^N}{|u\left(y\right)|}^2{\gamma }_k\left(dy\right).}$$

(79)

Proof. 

From Equations (65), (67), (72), (78) and Fubini’s theorem, we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{|{{\rm Y}}_{\varphi }^M\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)}\hfill \\ & =& {\displaystyle {|b|}^{2\langle k\rangle +N}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}u\right)\left(\lambda \right)\right|}^2{|{\mathfrak{F}}_D^M\left(\varphi \right)\left(\lambda r\right)|}^2{\gamma }_k\left(d\lambda \right)\frac{dr}{r}}\hfill \\ & =& {\displaystyle {|b|}^{2\langle k\rangle +N}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}u\right)\left(\lambda \right)\right|}^2\left({\int }_0^{\infty }{\left|{\mathfrak{F}}_D^M\left(\varphi \right)\left(\lambda r\right)\right|}^2\frac{dr}{r}\right){\gamma }_k\left(d\lambda \right).}\hfill \end{array}$$

Using Equations (56) and (68) we derive

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{{\rm Y}}_{\varphi }^M{\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)=c(b,k,\varphi ){∥{\mathbf{L}}_{\frac{-2a}{b}}u∥}_{L_k^2\left({\mathbb{R}}^N\right)}=C_{\varphi }^M{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.}\end{array}$$

□

Theorem 3

(Orthogonality Property). Let ϕ be an admissible Dunkl linear canonical wavelet. Then for all $f_1,f_2\in L_k^2\left({\mathbb{R}}^N\right)$ we have the identity

$${\displaystyle {\int }_{{\mathbb{R}}^N}{f}_1\left(y\right)\overline{f_2\left(y\right)}{\gamma }_k\left(dy\right)=\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }{{\rm Y}}_{\varphi }^M\left(f_1\right)(r,t)\overline{{{\rm Y}}_{\varphi }^M\left(f_2\right)(r,t)}d{\mu }_k(r,t).}$$

(80)

Proof. 

The Plancherel’s formula for DLCWT implies that for all $f\in L_k^2\left({\mathbb{R}}^N\right)$, the function

$$I\left(r\right):={\left({\int }_{{\mathbb{R}}^N}{|{{\rm Y}}_{\varphi }^M\left(f\right)(r,t)|}^2{\gamma }_k\left(dt\right)\right)}^{1/2}$$

is in $L_k^2\left({\mathbb{R}}^N\right)$. Consequently, ${{\rm Y}}_{\varphi }^M\left(f\right)(r,·)\in L_k^2\left({\mathbb{R}}^N\right)$ for almost all $r\in {\mathbb{R}}_{+}$. Using Equation (67) applied to the variable t, we obtain

$$\begin{array}{ccc}& & {\int }_{\Omega }{{\rm Y}}_{\varphi }^M\left(f_1\right)(r,t)\overline{{{\rm Y}}_{\varphi }^M\left(f_2\right)(r,t)}d{\mu }_k(r,t)\hfill \\ & & ={\int }_{\Omega }{r}^{2\langle k\rangle +N}\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\underset{M^{-1}}{\odot }{\varphi }_r^M\right)\left(t\right)\overline{\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\underset{M^{-1}}{\odot }{\varphi }_r^M\right)\left(t\right)}d{\mu }_k(r,t)\hfill \\ & & ={|b|}^{2\langle k\rangle +N}{\int }_{\Omega }{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}{|{\mathfrak{F}}_D^M\left(\varphi \right)\left(r\lambda \right)|}^2{\gamma }_k\left(d\lambda \right){\displaystyle \frac{dr}{r}}.\hfill \end{array}$$

In view of Fubini’s theorem, we can justify the interchanging of the integral

$$\begin{array}{ccc}& & {|b|}^{2\langle k\rangle +N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}\left({\displaystyle {\int }_0^{\infty }{|{\mathfrak{F}}_D^M\left(\varphi \right)\left(r\lambda \right)|}^2\frac{dr}{r}}\right){\gamma }_k\left(d\lambda \right)\hfill \\ & & =c(b,k,\varphi ){\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}{\gamma }_k\left(d\lambda \right).\hfill \end{array}$$

The conclusion follows due to Equation (57). □

Proposition 4.

${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is a reproducing kernel Hilbert space (RKHS) with kernel function

$${\displaystyle {\mathcal{K}}_{\varphi }^M(r^{\prime },x^{\prime };r,x):=\frac{1}{c(b,k,\varphi )}{\int }_{{\mathbb{R}}^N}{\varphi }_{r^{\prime },x^{\prime }}^M\left(y\right)\overline{{\varphi }_{r,x}^M\left(y\right)}{\gamma }_k\left(dy\right),}$$

(81)

which satisfies

$$\forall (r^{\prime },x^{\prime }),(r,x)\in \Omega ,\left|{\mathcal{K}}_{\varphi }^M(r^{\prime },x^{\prime };r,x)\right|\le {\displaystyle \frac{{\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{c(b,k,\varphi )}}.$$

(82)

Proof. 

Let f be in $L_k^2\left({\mathbb{R}}^N\right)$. We have

$${\displaystyle {{\rm Y}}_{\varphi }^M\left(f\right)(r,x)={\int }_{{\mathbb{R}}^N}f\left(y\right)\overline{{\varphi }_{r,x}^M\left(y\right)}{\gamma }_k\left(dy\right),(r,x)\in \Omega .}$$

Using Equation (80), we obtain

$${\displaystyle {{\rm Y}}_{\varphi }^M\left(f\right)(r,x)=\frac{1}{c(b,k,\varphi )}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\varphi }^M\left(f\right)(r^{\prime },x^{\prime })\overline{{{\rm Y}}_{\varphi }^M\left({\varphi }_{r,x}^M\right)(r^{\prime },x^{\prime })}d{\mu }_k(r^{\prime },x^{\prime }).}$$

On the other hand, using Equation (67), one can easily see that the function

$${\displaystyle x^{\prime }\mapsto \frac{1}{c(b,k,\varphi )}\overline{{{\rm Y}}_{\varphi }^M\left({\varphi }_{r,x}^M\right)}(r^{\prime },x^{\prime })=\frac{1}{c(b,k,\varphi )}{\int }_{{\mathbb{R}}^N}{\varphi }_{r^{\prime },x^{\prime }}^M\left(y\right)\overline{{\varphi }_{r,x}^M\left(y\right)}{\gamma }_k\left(dy\right)}$$

belongs to $L_k^2\left({\mathbb{R}}^N\right)$, for every $(r,x),(r^{\prime },x^{\prime })\in \Omega$. Therefore, the result is obtained. □

2.2. Toeplitz Operators

Let $\varsigma$ be measurable function on $\Omega$, we define the Toeplitz operator on $L_k^2\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(u\right)\left(y\right)=\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }\varsigma (r,t){{\rm Y}}_{\varphi }^M\left(u\right)(r,t){\varphi }_{r,t}^M\left(y\right)d{\mu }_k(r,t),y\in {\mathbb{R}}^N.}$$

(83)

In this section, $\varphi$ will be a an admissible linear canonical Dunkl wavelet on ${\mathbb{R}}^N$ such that ${\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$ The boundedness and compactness for Toeplitz operators and their properties has been studied in Ref. [22] in more general settings. In this section, we simply give the proofs for the results that we will use in this paper for the sake of completeness.

Proposition 5.

If $\varsigma \in L_{{\mu }_k}^1(\Omega )$, then ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ belongs to $S_{\infty }$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_{\infty }}⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )},$$

(84)

where $S_{\infty }:=B\left(L_k^2\left({\mathbb{R}}^N\right)\right)$.

Proof. 

For every functions u and v in $L_k^2\left({\mathbb{R}}^N\right)$, we have from Equations (18) and (77),

$$\begin{array}{cc}\hfill |{\langle {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(u\right),v\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}|& {\displaystyle ⩽\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }|\varsigma \left(z\right)||{{\rm Y}}_{\varphi }^M\left(u\right)\left(z\right)||\overline{{{\rm Y}}_{\varphi }^M\left(v\right)\left(z\right)}|d{\mu }_k\left(z\right)}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}\parallel {{\rm Y}}_{\varphi }^M{\left(u\right)\parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}\parallel {{\rm Y}}_{\varphi }^M{\left(v\right)\parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Thus we have the desired result. □

Proposition 6.

If $\varsigma \in L_{{\mu }_k}^{\infty }(\Omega )$, then ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ belongs to $S_{\infty }$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_{\infty }}⩽{\parallel \varsigma \parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}.$$

(85)

Proof. 

Let $u,v\in L_k^2\left({\mathbb{R}}^N\right)$. Then

$$\begin{array}{cc}\hfill |{\langle {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(u\right),v\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}|& {\displaystyle ⩽\frac{1}{c(b,k,\varphi )}{\int }_{\Omega }|\varsigma \left(z\right)||{{\rm Y}}_{\varphi }^M\left(u\right)\left(z\right)||\overline{{{\rm Y}}_{\varphi }^M\left(v\right)\left(z\right)}|d{\mu }_k\left(z\right)}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}\parallel {{\rm Y}}_{\varphi }^M{\left(u\right)\parallel }_{L_{{\mu }_k}^2(\Omega )}{\parallel {{\rm Y}}_{\varphi }^M\left(v\right)\parallel }_{L_{{\mu }_k}^2(\Omega )}.\hfill \end{array}$$

By Equation (79), we have

$$\begin{array}{c}\hfill |{\langle {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(u\right),v\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}{|⩽\parallel u\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}\end{array}$$

as desired. □

Therefore, by Equations (84), (85) and the Riesz–Thorin interpolation theorem (see [29] (Theorem 2) and [30] (Theorem 2.11), we deduce the result.

Theorem 4.

Let $1\le p\le \infty$. If ς belongs to $L_{{\mu }_k}^p(\Omega )$, there is a unique linear bounded operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$, satisfying

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_{\infty }}{⩽\left(c(b,k,\varphi )\right)}^{-{\displaystyle \frac{1}{p}}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^p(\Omega )}.$$

(86)

Now we will prove that the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ belongs to the Schatten class $S_p$. We start with the result.

Proposition 7.

If $\varsigma \in L_{{\mu }_k}^1(\Omega )$, then the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is in $S_2$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_2}⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}.$$

(87)

Proof. 

Given ${\left\{h_j\right\}}_{j\in \mathbb{N}}$ an orthonormal basis for $L_k^2\left({\mathbb{R}}^N\right)$. From Parseval Equality and Equations (18), (19), (74), we have

$$\begin{array}{ccc}& & \sum _{j=1}^{\infty }{∥{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right)∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2=\sum _{j=1}^{\infty }{〈{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right),{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right)〉}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & =& {\displaystyle \frac{1}{c(b,k,\varphi )}\sum _{j=1}^{\infty }{\int }_{\Omega }\varsigma (r,t){〈h_j,{\varphi }_{r,t}^M〉}_{L_k^2\left({\mathbb{R}}^N\right)}{\overline{〈{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right),{\varphi }_{r,t}^M〉}}_{L_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,t)}\hfill \\ & =& {\displaystyle \frac{1}{c(b,k,\varphi )}{\int }_{\Omega }\varsigma (r,t)\sum _{j=1}^{\infty }{〈h_j,{\varphi }_{r,t}^M〉}_{L_k^2\left({\mathbb{R}}^N\right)}{〈({\mathfrak{T}}_{\varphi }^M{\left(\varsigma )\right)}^{*}{\varphi }_{r,t}^M,h_j〉}_{L_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,t)}\hfill \\ & =& {\displaystyle \frac{1}{c(b,k,\varphi )}{\int }_{\Omega }\varsigma (r,t){〈({\mathfrak{T}}_{\varphi }^M{\left(\varsigma )\right)}^{*}{\varphi }_{r,t}^M,{\varphi }_{r,t}^M〉}_{L_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,t).}\hfill \end{array}$$

Therefore, from Equations (19), (71) and (84) we obtain

$$\begin{array}{ccc}\hfill \sum _{j=1}^{\infty }{\parallel {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2& \le & {\displaystyle \frac{1}{c(b,k,\varphi )}{\int }_{\Omega }|\varsigma (r,t)|\parallel {\left({\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\right)}^{*}{\parallel }_{S_{\infty }}d{\mu }_k(r,t)}\hfill \\ & \le & {\displaystyle \frac{1}{c{(b,k,\varphi )}^2}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}^2<\infty .\hfill \end{array}$$

Thus by Proposition 2.8 in the book [30], the operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is in $S_2$ and hence compact. □

Proposition 8.

If $\varsigma \in L_{{\mu }_k}^p(\Omega ),$$1⩽p<\infty$, then the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is compact.

Proof. 

Let $\varsigma$ be in $L_{{\mu }_k}^p(\Omega )$ and let ${\left({\varsigma }_j\right)}_{j\in \mathbb{N}}$ be a sequence of functions in $L_{{\mu }_k}^1(\Omega )\cap L_{{\mu }_k}^{\infty }(\Omega )$ such that ${\varsigma }_j\to \varsigma$ in $L_{{\mu }_k}^p(\Omega )$. It follows by Equation (86)

$$\parallel {\mathfrak{T}}_{\varphi }^M\left({\varsigma }_j\right)-{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right){\parallel }_{S_{\infty }}\le {\left(c(b,k.\varphi )\right)}^{-{\displaystyle \frac{1}{p}}}{\parallel {\varsigma }_j-\varsigma \parallel }_{L_{{\mu }_k}^p(\Omega )}.$$

(88)

Therefore ${\mathfrak{T}}_{\varphi }^M\left({\varsigma }_j\right)\to {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ in $S_{\infty }$ as $j\to \infty$. On the other hand, since ${\mathfrak{T}}_{\varphi }^M\left({\varsigma }_j\right)$ is in $S_2$ hence compact, it follows that ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is compact. □

Theorem 5. 

If $\varsigma \in L_{{\mu }_k}^1(\Omega )$, then the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is in $S_1$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_1}⩽{\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}.$$

(89)

Proof. 

Since $\varsigma$ is in $L_{{\mu }_k}^1(\Omega )$, then ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ is in $S_2$. Therefore from [30] (Theorem 2.2), there exists an orthonormal basis $\{h_j,j=1,2...\}$ for the orthogonal complement of the kernel of the operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$, consisting of eigenvectors of $|{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)|$ and $\{{\phi }_j,j=1,2,\dots \}$ an orthonormal set in $L_k^2\left({\mathbb{R}}^N\right)$, such that

$${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(u\right)=\sum _{j=1}^{\infty }{s}_j{\langle f,h_j\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}{\phi }_j,$$

(90)

where $s_j,j=1,2...$ are the positive singular values of ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)$ corresponding to $h_j$. Then

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_1}=\sum _{j=1}^{\infty }{s}_j=\sum _{j=1}^{\infty }{〈{\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right),{\phi }_j〉}_{L_k^2\left({\mathbb{R}}^N\right)}.$$

Thus, by Bessel inequality and Equations (71) and (74), we have

$$\begin{array}{ccc}\hfill \parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_1}& =& \sum _{j=1}^{\infty }{\langle {\mathfrak{T}}_{\varphi }^M\left(\varsigma \right)\left(h_j\right),{\phi }_j\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & =& {\displaystyle \frac{1}{c(b,k,\varphi )}}\sum _{j=1}^{\infty }{\int }_{\Omega }\varsigma (r,t){{\rm Y}}_{\varphi }^M\left(h_j\right)(r,t)\overline{{{\rm Y}}_{\varphi }^M\left({\phi }_j\right)(r,t)}d{\mu }_k(r,t)\hfill \\ & \le & {\displaystyle \frac{1}{c(b,k,\varphi )}}{\int }_{\Omega }|\varsigma (r,t)|{\left(\sum _{j=1}^{\infty }{|{{\rm Y}}_{\varphi }^M\left(h_j\right)(r,t)|}^2\right)}^{{\displaystyle \frac{1}{2}}}{\left(\sum _{j=1}^{\infty }{|{{\rm Y}}_{\varphi }^M\left({\phi }_j\right)(r,t)|}^2\right)}^{{\displaystyle \frac{1}{2}}}d{\mu }_k(r,t)\hfill \\ & \le & {\displaystyle \frac{1}{c(b,k,\varphi )}}{\int }_{\Omega }|\varsigma (r,t)|\parallel {\varphi }_{r,t}^M{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{2}d{\mu }_k(r,t)\hfill \\ & ⩽& {\displaystyle \frac{1}{c(b,k,\varphi )}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

□

Thus from Equation (85), Equation (89) and by interpolation (See [30], Theorem 2.10 and Theorem 2.11) we deduce the result.

Corollary 1. 

If $\varsigma \in L_{{\mu }_k}^p(\Omega )$, $1⩽p⩽\infty$, then the Toeplitz operator ${\mathfrak{T}}_{\varphi }^M\left(\varsigma \right):L_k^2\left({\mathbb{R}}^N\right)⟶L_k^2\left({\mathbb{R}}^N\right)$ is in $S_p$, such that

$$\parallel {\mathfrak{T}}_{\varphi }^M{\left(\varsigma \right)\parallel }_{S_p}⩽{\left(c(b,k,\varphi )\right)}^{-{\displaystyle \frac{1}{p}}}{\parallel \varsigma \parallel }_{L_{{\mu }_k}^p(\Omega )}.$$

3. Spectral Analysis of Concentration Operators and Main Results

Throughout this section $\varphi$ will be a linear canonical Dunkl wavelet in $L_k^2\left({\mathbb{R}}^N\right)$ such that ${∥\varphi ∥}_{L_k^2\left({\mathbb{R}}^N\right)}=1$.

In this section we shall keep our focus on the deformed concentration operators noted by ${\mathcal{L}}_{\varphi }^M\left(R\right)$ and defined via the Toeplitz operators by

$${\mathcal{L}}_{\varphi }^M\left(R\right):={\mathfrak{T}}_{\varphi }^M\left(\varsigma \right),$$

(91)

where $\varsigma ={\chi }_R$ and R is a subset of $\Omega$ with finite measure $0<{\mu }_k\left(R\right)<\infty .$

Using the fact that ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is a reproducing kernel Hilbert space (RKHS), we will define a Calderón–Toeplitz-type operator, and study its boundedness and compactness by giving a trace formulas. Then we will show some uncertainty principles for essentially localized functions. Finally we will introduce the scalogram and show how a finite vector space spanned by the first eigenfunctions of the concentration operator can effectively approximate functions that are essentially localized in subsets of finite measure, and corresponding error estimates are given.

3.1. Range of the DLCWT and Calderón–Toeplitz-Type Operator

We have $P_{\varphi }^M$ is the integral operator

$${\displaystyle P_{\varphi }^{M}F\left(z\right)={\int }_{\Omega }F(r,x){\mathcal{K}}_{\varphi }^M(z;r,x)d{\mu }_k(r,x),z=(r^{\prime },x^{\prime })\in \Omega ,}$$

(92)

with integral kernel ${\mathcal{K}}_{\varphi }^M$ given by Equation (81).

Since ${\mathcal{K}}_{\varphi }^M$ is the integral kernel of an orthogonal projection, then it satisfies

$${\mathcal{K}}_{\varphi }^M(z;z^{\prime })=\overline{{\mathcal{K}}_{\varphi }^M(z^{\prime };z)},\forall z,z^{\prime }\in \Omega ,$$

(93)

and for all $z,z^{\prime }\in \Omega$,

$${\displaystyle {\mathcal{K}}_{\varphi }^M(z;z^{\prime })={\int }_{\Omega }{\mathcal{K}}_{\varphi }^M(z;z^{″}){\mathcal{K}}_{\varphi }^M(z^{″};z^{\prime })d{\mu }_k\left(z^{″}\right).}$$

(94)

Moreover, if ${\left\{{\theta }_j\right\}}_{j\in \mathbb{N}}$ is any orthonormal basis for ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$, then

$${\mathcal{K}}_{\varphi }^M(z;z^{\prime })=\sum _{j=1}^{\infty }{\theta }_j\left(z\right)\overline{{\theta }_j\left(z^{\prime }\right)}.$$

(95)

Definition 5. 

Let ${\mathfrak{C}}_{\varphi ,R,M}:{{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\to {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ the operator (a Calderón–Toeplitz-type operator) defined by

$${\mathfrak{C}}_{\varphi ,R,M}F=P_{\varphi }^M{P}_{R}F.$$

(96)

Proposition 9. 

The operator ${\mathfrak{C}}_{\varphi ,R,M}:{{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\to {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is nonnegative and bounded, such that

$${\mathfrak{C}}_{\varphi ,R,M}={{\rm Y}}_{\varphi }^M{\mathcal{L}}_{\varphi }^M\left(R\right){\left({{\rm Y}}_{\varphi }^M\right)}^{*}$$

(97)

and

$$0\le {\mathfrak{C}}_{\varphi ,R,M}\le P_R\le I.$$

(98)

Proof. 

Let H be a function in ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$. Then

$$\begin{array}{ccc}\hfill {〈{\mathfrak{C}}_{\varphi ,R,M}H,H〉}_{L_{{\mu }_k}^2(\Omega )}& =& {〈P_{\varphi }^M\left(P_{R}H\right),H〉}_{L_{{\mu }_k}^2(\Omega )}\hfill \\ & =& {\displaystyle {〈P_{R}H,H〉}_{L_{{\mu }_k}^2(\Omega )}={\int }_R{|H(r,x)|}^{2}d{\mu }_k(r,x).}\hfill \end{array}$$

(99)

Thus we deduce Equation (98), and ${\mathfrak{C}}_{\varphi ,R,M}$ is bounded and positive.

Now, we want to prove Equation (97). Indeed, using ${{\rm Y}}_{\varphi }^M$ and ${\left({{\rm Y}}_{\varphi }^M\right)}^{*}$, the time-frequency Toeplitz operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ satisfies

$${\mathcal{L}}_{\varphi }^M\left(R\right)\left(h\right)={\left({{\rm Y}}_{\varphi }^M\right)}^{*}\left(P_R{{\rm Y}}_{\varphi }^{M}h\right),h\in L_k^2\left({\mathbb{R}}^N\right).$$

Then

$$\left({{\rm Y}}_{\varphi }^M{\mathcal{L}}_{\varphi }^M\left(R\right){\left({{\rm Y}}_{\varphi }^M\right)}^{*}\right)H=P_{\varphi }^M{P}_{R}H={\mathfrak{C}}_{\varphi ,R,M}H,H\in {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right).$$

(100)

Therefore the time-frequency operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ and the deformed Calderón–Toeplitz operator ${\mathfrak{C}}_{\varphi ,R,M}$ are related by

$${\mathfrak{C}}_{\varphi ,R,M}={{\rm Y}}_{\varphi }^M{\mathcal{L}}_{\varphi }^M\left(R\right){\left({{\rm Y}}_{\varphi }^M\right)}^{*}.$$

□

From the above proposition we deduce that ${\mathfrak{C}}_{\varphi ,R,M}$ and ${\mathcal{L}}_{\varphi }^M\left(R\right)$ enjoy the same spectral properties, in particular, we have this proposition.

Proposition 10. 

The deformed Calderón–Toeplitz operator ${\mathfrak{C}}_{\varphi ,R,M}$ is compact and even trace class with

$$tr\left({\mathfrak{C}}_{\varphi ,R,M}\right)=tr\left({\mathcal{L}}_{\varphi }^M\left(R\right)\right)={\mathfrak{R}}_k(\varphi ,R,M),$$

(101)

where

$${\displaystyle {\mathfrak{R}}_k(\varphi ,R,M):={\int }_R{\parallel {\varphi }_{r,x}^M\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{2}d{\mu }_k(r,x).}$$

(102)

Proof.  

Let ${\left\{v_i\right\}}_{i=1}^{\infty }$ be any orthonormal basis for ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right).$ Then if we denote $u_j$ by $u_i=\sqrt{c(b,k,\varphi )}{\left({{\rm Y}}_{\varphi }^M\right)}^{*}\left(v_i\right),$ then ${\left\{u_i\right\}}_{j=1}^{\infty }$ is an orthonormal basis for $L_k^2\left({\mathbb{R}}^N\right)$.

Thus by Equation (18) and Fubini’s theorem

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{i=1}^{\infty }{〈{\mathfrak{C}}_{\varphi ,R,M}\left(v_i\right),v_i〉}_{L_{{\mu }_k}^2(\Omega )}}& =& {\displaystyle \sum _{i=1}^{\infty }{〈{\mathcal{L}}_{\varphi }^M\left(R\right){\left({{\rm Y}}_{\varphi }^M\right)}^{*}\left(v_i\right),{\left({{\rm Y}}_{\varphi }^M\right)}^{*}\left(v_i\right)〉}_{L_k^2\left({\mathbb{R}}^N\right)}}\hfill \\ & =& {\displaystyle \sum _{i=1}^{\infty }{\int }_R{\left|{{\rm Y}}_{\varphi }^M\left(u_i\right)\right|}^{2}d{\mu }_k}\hfill \\ & =& {\displaystyle {\int }_R\sum _{i=1}^{\infty }{\left|{{\rm Y}}_{\varphi }^M\left(u_j\right)\right|}^{2}d{\mu }_k}\hfill \\ & =& {\displaystyle {\int }_R\sum _{i=1}^{\infty }{\left|{〈u_i,{\varphi }_{r,x}^M〉}_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^{2}d{\mu }_k(r,x)}\hfill \\ & =& {\displaystyle {\int }_R{∥{\varphi }_{r,x}^M∥}_{L_k^2\left({\mathbb{R}}^N\right)}^{2}d{\mu }_k(r,x)}\hfill \\ & =& {\mathfrak{R}}_k(\varphi ,R,M).\hfill \end{array}$$

Therefore, the operator ${\mathfrak{C}}_{\varphi ,R,M}$ is trace class with $\parallel {\mathfrak{C}}_{\varphi ,R,M}{\parallel }_{S_1}=tr\left({\mathfrak{C}}_{\varphi ,R,M}\right)={\mathfrak{R}}_k(\varphi ,R,M).$ □

We define the operator ${\mathcal{V}}_{\varphi ,R,M}:L_k^2\left({\mathbb{R}}^N\right)\to L_k^2\left({\mathbb{R}}^N\right)$ by ${\mathcal{V}}_{\varphi ,R,M}=P_{\varphi }^M{P}_R{P}_{\varphi }^M.$ The advantage of ${\mathcal{V}}_{\varphi ,R,M}$ compared to ${\mathfrak{C}}_{\varphi ,R,M}$ is that it is defined on $L_k^2\left({\mathbb{R}}^N\right)$ and consequently its spectral properties can be easily related to its integral kernel. Since ${\mathfrak{C}}_{\varphi ,R,M}$ is positive and trace-class, then using the decomposition

$$L_{{\mu }_k}^2(\Omega )={{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\oplus {\left({{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\right)}^{\perp },$$

we deduce that ${\mathcal{V}}_{\varphi ,R,M}$ is also positive and trace-class with

$$tr\left({\mathcal{V}}_{\varphi ,R,M}\right)=tr\left({\mathfrak{C}}_{\varphi ,R,M}\right)={\mathfrak{R}}_k(\varphi ,R,M).$$

(103)

Proposition 11. 

The trace of the operator ${\mathfrak{C}}_{\varphi ,R,M}^2$ is provided by

$${\displaystyle tr\left({\mathfrak{C}}_{\varphi ,R,M}^2\right)={\int }_R{\int }_R{|{\mathcal{K}}_{\varphi }^M(z;z^{\prime })|}^{2}d{\mu }_k\left(z\right)d{\mu }_k\left(z^{\prime }\right).}$$

(104)

Proof. 

As ${\mathcal{V}}_{\varphi ,R,M}$ is positive, then

$$tr\left({\mathfrak{C}}_{\varphi ,R,M}^2\right)=tr\left({\mathcal{V}}_{\varphi ,R,M}^2\right).$$

(105)

Now, since ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is a RKHS, we have for all $F\in {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$,

$${\displaystyle {\mathcal{V}}_{\varphi ,R,M}F(x,t)={\int }_{\Omega }F(x^{\prime },t^{\prime }){\int }_{\Omega }{\chi }_R(s,y){\mathcal{K}}_{\varphi }^M(x,t;s,y){\mathcal{K}}_{\varphi }^M(s,y;x^{\prime },t^{\prime })d{\mu }_k(s,y)d{\mu }_k(x^{\prime },t^{\prime }).}$$

That is, ${\mathcal{V}}_{\varphi ,R,M}$ has integral kernel

$${\displaystyle {\mathcal{N}}_{\varphi ,R}^M(x,t;x^{\prime },t^{\prime })={\int }_{\Omega }{\chi }_R(s,y){\mathcal{K}}_{\varphi }^M(x,t;s,y){\mathcal{K}}_{\varphi }^M(s,y;x^{\prime },t^{\prime })d{\mu }_k(s,y).}$$

(106)

Therefore

$$\begin{array}{ccc}\hfill tr\left({\mathcal{V}}_{\varphi ,R,M}^2\right)& =& {\displaystyle {\int }_{\Omega }{\int }_{\Omega }{|{\mathcal{N}}_{\varphi ,R}^M(z;z^{\prime })|}^{2}d{\mu }_k\left(z\right)d{\mu }_k\left(z^{\prime }\right)}\hfill \\ & =& {\displaystyle {\int }_{\Omega }{\int }_{\Omega }{\mathcal{N}}_{\varphi ,R}^M(x,t;x^{\prime },t^{\prime })\overline{{\mathcal{N}}_{\varphi ,R}^M(x,t;x^{\prime },t^{\prime })}d{\mu }_k(x,t)d{\mu }_k(x^{\prime },t^{\prime })}\hfill \\ & =& {\displaystyle {\int }_{\Omega }{\int }_{\Omega }{\chi }_R\left(z_1\right){\chi }_R\left(z_2\right){\mathbf{K}}_{\varphi }^M(z_1;z_2)d{\mu }_k\left(z_1\right)d{\mu }_k\left(z_2\right),}\hfill \end{array}$$

where by using the properties of the kernel of the reproducing kernel Hilbert space

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\varphi }^M(z_1;z_2)& =& {\displaystyle {\int }_{\Omega }{\int }_{\Omega }{\mathcal{K}}_{\varphi }^M(z_2;x,t){\mathcal{K}}_{\varphi }^M(x,t;z_1){\mathcal{K}}_{\varphi }^M(z_1;x^{\prime },t^{\prime }){\mathcal{K}}_{\varphi }^M(x^{\prime },t^{\prime };z_2)d{\mu }_k(x,t)d{\mu }_k(x^{\prime },t^{\prime })}\hfill \\ & =& {\mathcal{K}}_{\varphi }^M(z_2;z_1){\mathcal{K}}_{\varphi }^M(z_1;z_2).\hfill \end{array}$$

Using Equation (94), we get

$${\mathbf{K}}_{\varphi }^M(z_1;z_2)={|{\mathcal{K}}_{\varphi }^M(z_1;z_2)|}^2.$$

(107)

This follows us to conclude. □

By Equations (35) and (97), we can deduce that the Calderón–Toeplitz-type operator ${\mathfrak{C}}_{\varphi ,R,M}:{{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)\to {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ is diagonalizable as

$${\displaystyle {\mathfrak{C}}_{\varphi ,R,M}F=\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){〈F,t_j^R〉}_{L_{{\mu }_k}^2(\Omega )}{t}_j^R,F\in {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right),}$$

(108)

where $t_j^R={\displaystyle \frac{1}{\sqrt{c(b,k,\varphi )}}}{{\rm Y}}_{\varphi }^M\left({\psi }_j^R\right)$. Then we have the result.

Lemma 1. 

For all $z=(x,t)\in \Omega ,$ we have

$${\displaystyle {\Theta }_k^M\left(z\right):={\int }_{\Omega }{\chi }_R\left(\omega \right){|{\mathcal{K}}_{\varphi }^M(\omega ;z)|}^{2}d{\mu }_k\left(\omega \right)=\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){\mathbf{S}}_{\varphi ,M}^D\left({\psi }_j^R\right)\left(z\right).}$$

(109)

Proof. 

By Equation (81), ${\mathcal{K}}_{\varphi }^M(·;z)$ belongs to ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$, for all $z=(x,t)\in \Omega$. It follows that

$$\begin{array}{ccc}\hfill {〈{\mathfrak{C}}_{\varphi ,R,M}{\mathcal{K}}_{\varphi }^M(·;z),{\mathcal{K}}_{\varphi }^M(·;z)〉}_{L_{{\mu }_k}^2(\Omega )}& =& {〈P_{\varphi }^M{P}_R{\mathcal{K}}_{\varphi }^M(·;z),{\mathcal{K}}_{\varphi }^M(·;z)〉}_{L_{{\mu }_k}^2(\Omega )}\hfill \\ & =& {\displaystyle {\int }_{\Omega }{\chi }_R\left(\omega \right){\mathcal{K}}_{\varphi }^M(\omega ;z)\overline{{\mathcal{K}}_{\varphi }^M(\omega ;z)}d{\mu }_k\left(\omega \right)}\hfill \\ & =& {\displaystyle {\int }_{\Omega }{\chi }_R(·){|{\mathcal{K}}_{\varphi }^M(·;z)|}^{2}d{\mu }_k(·).}\hfill \end{array}$$

Let ${\left\{w_j^R\right\}}_{j=1}^{\infty }\subset {{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ be an orthonormal basis of $\mathrm{Ker}\left({\mathfrak{C}}_{\varphi ,R,M}\right)$ (eventually empty). Then ${\left\{t_j^R\right\}}_{j=1}^{\infty }\cup {\left\{w_j^R\right\}}_{j=1}^{\infty }$ is an orthonormal basis of ${{\rm Y}}_{\varphi }^M\left(L_k^2\left({\mathbb{R}}^N\right)\right)$ and therefore the reproducing kernel ${\mathcal{K}}_{\varphi }^M$ can be written as

$${\displaystyle {\mathcal{K}}_{\varphi }^M(x,t;x^{\prime },t^{\prime })=\overline{{\mathcal{K}}_{\varphi }^M(x^{\prime },t^{\prime };z)}=\sum _{i=1}^{\infty }{w}_i^R\left(z\right)\overline{w_i^R(x^{\prime },t^{\prime })}+\sum _{i=1}^{\infty }{t}_i^R\left(z\right)\overline{t_i^R(x^{\prime },t^{\prime })}.}$$

(110)

Using this, we compute again

$$\begin{array}{ccc}{〈{\mathfrak{C}}_{\varphi ,R,M}{\mathcal{K}}_{\varphi }^M(·;z),{\mathcal{K}}_{\varphi }^M(·;z)〉}_{L_{{\mu }_k}^2(\Omega )}\hfill & =\hfill & {〈{\displaystyle {\mathfrak{C}}_{\varphi ,R,M}\sum _{i=1}^{\infty }\overline{t_i^R\left(z\right)}{t}_i^R,\sum _{m=1}^{\infty }\overline{t_m^R\left(z\right)}{t}_m^R}〉}_{L_{{\mu }_k}^2(\Omega )}\hfill \\ & =\hfill & {\displaystyle \sum _{i,m}\overline{t_i^R\left(z\right)}{t}_m^R\left(z\right){〈{\mathfrak{C}}_{\varphi ,R,M}{t}_i^R,t_m^R〉}_{L_{{\mu }_k}^2(\Omega )}}\hfill \\ & =\hfill & {\displaystyle \sum _{i=1}^{\infty }{\alpha }_i\left(R\right){|t_i^R\left(z\right)|}^2,}\hfill \end{array}$$

and the conclusion follows. □

3.2. Uncertainty Relations

For $0\le \epsilon <1$, we say that a nonzero function $f\in L_k^2\left({\mathbb{R}}^N\right)$ is $(\epsilon ,\varphi )$-concentrated in R if

$${∥P_{R^c}{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\le c(b,k,\varphi )\epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(111)

where $R^c={\mathbb{R}}^N\setminus R$. Notice that, if $\epsilon =0$ in Equation (111), then we say that f is concentrated in R, and in the case of $0<\epsilon <1$, the subset R will be called the essential support of ${{\rm Y}}_{\varphi }^M\left(f\right)$.

Let ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ be the subset of $L_k^2\left({\mathbb{R}}^N\right)$ containing all functions that satisfy Equation (111). Therefore we obtain the Donoho-Stark-type uncertainty principle for the DLCWT.

Theorem 6. 

If $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then R satisfies

$${\mu }_k\left(R\right)\ge c(b,k,\varphi )(1-\epsilon ).$$

(112)

Proof. 

We have by Equation (79)

$${c(b,k,\varphi )\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2={∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2+{∥{\chi }_{R^c}{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2.$$

Then

$${∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\ge c(b,k,\varphi ){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2(1-\epsilon ).$$

On the other hand by Equation (77)

$${∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\le {\mu }_k\left(R\right){∥{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^{\infty }(\Omega )}^2\le {\mu }_k\left(R\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

Therefore we obtain the desired result. □

Definition 6. 

We say that $f\in L_k^2\left({\mathbb{R}}^N\right)\setminus \left\{0\right\}$ is ε-localized with respect to an operator A if

$${\parallel Af-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\le \epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(113)

Note that, if $\epsilon =0$ in Equation (113), then f is an eigenvector of the operator A corresponding to the eigenvalue 1.

Proposition 12. 

Let $f\in L_k^2\left({\mathbb{R}}^N\right)$. Then $f\in {\mathcal{C}}_{\epsilon ,\phi }\left(R\right)$ if, and only if,

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right),f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\ge (1-\epsilon ){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(114)

Proof. 

Since the operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ is compact and self-adjoint, then by Equations (79) and (111), we have

$$\begin{array}{ccc}\hfill {\langle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right),f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}& =& {\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_R{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\hfill \\ & =& {\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2-{\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_{R^c}{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2\hfill \\ & =& {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\displaystyle \frac{1}{c(b,k,\varphi )}}{∥{\chi }_{R^c}{{\rm Y}}_{\varphi }^M\left(f\right)∥}_{L_{{\mu }_k}^2\left(\Omega \right)}^2,\hfill \end{array}$$

and the result follows. □

Equation (114) is equivalent to

$${〈(I-{\mathcal{L}}_{\varphi }^M\left(R\right))f,f〉}_{L_k^2\left({\mathbb{R}}^N\right)}\le \epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(115)

Moreover, we have this comparison between $\epsilon$-concentration and $\epsilon$-localization.

Proposition 13. 

Let $f\in L_k^2\left({\mathbb{R}}^N\right)$.

1. 

If $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then f is ε-localized with respect to ${\mathcal{L}}_{\varphi }^M\left(R\right)$.

2. 

If f is ε-localized with respect to ${\mathcal{L}}_{\varphi }^M\left(R\right)$, then $f\in {\mathcal{C}}_{\epsilon +\sqrt{\epsilon },\varphi }\left(R\right)$.

Proof. 

The operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ is self-adjoint and by Equation (85), it is bounded with norm less than 1, then

$$\begin{array}{ccc}\hfill \parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2& =& {〈{\left(I-{\mathcal{L}}_{\varphi }^M\left(R\right)\right)}^{2}f,f〉}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \le & {〈\left(I-{\mathcal{L}}_{\varphi }^M\left(R\right)\right)f,f〉}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \le & {\epsilon \parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,\hfill \end{array}$$

we obtain then the first statement. Now, if f be $\epsilon$-localized with respect to ${\mathcal{L}}_{\varphi }^M\left(R\right)$, then

$$\begin{array}{ccc}& & 2{\langle (I-{\mathcal{L}}_{\varphi }^M\left(R\right))f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & =& \parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel {\mathcal{L}}_{\varphi }^M\left(R\right)f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & \le & \parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+{\left(\parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}+{\parallel {\mathcal{L}}_{\varphi }^M\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\right)}^2\hfill \\ & -& \parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & =& 2\parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+2\parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel {\mathcal{L}}_{\varphi }^M\left(R\right)f-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Thus,

$${\langle (I-{\mathcal{L}}_{\varphi }^M\left(R\right))f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\le \parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)\left(f\right)-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+\parallel {\mathcal{L}}_{\varphi }^M{\left(R\right)\left(f\right)-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)},$$

and we obtain the second statement. □

Finally we will state the Donoho–Stark-type uncertainty principle for concentration operators ${\mathcal{L}}_{\varphi }^M\left(R\right)$. In the remaining of this subsection $R_1,R_2\subset \Omega$ are subsets of finite measure and ${\varphi }_1,{\varphi }_2\in L_k^2\left({\mathbb{R}}^N\right)$ are admissible Dunkl linear canonical wavelets such that $\parallel {\varphi }_1{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel {\varphi }_2\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$

Theorem 7. 

Let ${\epsilon }_1,{\epsilon }_2\in (0,1)$, such that $\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}<1$. If $f\in L_k^2\left({\mathbb{R}}^N\right)$ is ${\epsilon }_1$-localized with respect to ${\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)$ and ${\epsilon }_2$-localized with respect to ${\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)$, then

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)\left(1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}\right).$$

(116)

Proof. 

We have by Equation (85),

$$\begin{array}{ccc}& & {∥f-{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right){\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \le & {∥f-{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)f-{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right){\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \le & {∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)f-f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)∥}_{S_{\infty }}{∥{\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)f-f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \le & (\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill {∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right){\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)f∥}_{L_k^2\left({\mathbb{R}}^N\right)}& \ge & {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}-{∥f-{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right){\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \ge & \left(1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Thus by Equation (84),

$$\begin{array}{ccc}\hfill 0<1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}& \le & {∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right){\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)∥}_{S_{\infty }}\hfill \\ & \le & {∥{\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)∥}_{S_{\infty }}{∥{\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)∥}_{S_{\infty }}\hfill \\ & \le & {\displaystyle \frac{{\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)}{c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)}}.\hfill \end{array}$$

The proof is complete. □

Remark 1.

1. 

Let ${\epsilon }_1,{\epsilon }_2\in (0,1)$ such that ${\epsilon }_1+\sqrt{{\epsilon }_1}<1$ and ${\epsilon }_2+\sqrt{{\epsilon }_2}<1$. If $f\in L_k^2\left({\mathbb{R}}^N\right)$ is ${\epsilon }_1$-localized with respect to ${\mathcal{L}}_{{\varphi }_1}^M\left(R_1\right)$ and ${\epsilon }_2$-localized with respect to ${\mathcal{L}}_{{\varphi }_2}^M\left(R_2\right)$, then by Theorem 6 and Proposition 13, we can immediately conclude that

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)(1-{\epsilon }_1-\sqrt{{\epsilon }_1})(1-{\epsilon }_2-\sqrt{{\epsilon }_2}).$$

(117)

2. 

Let ${\epsilon }_1,{\epsilon }_2\in (0,1)$, such that $\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}<1$. Then if $f\in {\mathcal{C}}_{{\epsilon }_1,{\varphi }_1}\left(R_1\right)\cap {\mathcal{C}}_{{\epsilon }_2,{\varphi }_2}\left(R_2\right)$, we have

$${\mu }_k\left(R_1\right){\mu }_k\left(R_2\right)\ge c(b,k,{\varphi }_1)c(b,k,{\varphi }_2)(1-\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}).$$

(118)

3.3. Scalogram of a Subspace

Given an n-dimensional subspace V of $L_k^2\left({\mathbb{R}}^N\right)$, $P_V$, the orthogonal projection onto V with projection kernel ${\mathfrak{G}}_V,$ i.e.,

$${\displaystyle P_{V}u(·)={\int }_{{\mathbb{R}}^N}{\mathfrak{G}}_V(·,t)u\left(t\right){\gamma }_k\left(dt\right).}$$

(119)

Recall that, if ${\left\{v_j\right\}}_{j=1}^n$ is an orthonormal basis of V, we have

$${\displaystyle {\mathfrak{G}}_V(r,x)=\sum _{j=1}^n{v}_j\left(r\right)\overline{v_j\left(x\right)}.}$$

(120)

The kernel ${\mathfrak{G}}_V$ is independent of the choice of orthonormal basis for V.

The scalogram of the space V with respect to $\varphi$ is defined by

$${\displaystyle {\mathbf{SCAL}}_{\varphi ,M}^{k}V(r,x):={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\mathfrak{G}}_V(y,z)\overline{{\varphi }_{r,x}^M\left(y\right)}{\varphi }_{r,x}^M\left(z\right){\gamma }_k\left(dy\right){\gamma }_k\left(dz\right),}$$

(121)

where ${\varphi }_{r,x}^M$ and ${\mathfrak{G}}_V$ are given in Equations (69) and (120).

Proposition 14. 

The scalogram ${\mathbf{SCAL}}_{\varphi ,M}^{k}V$ satisfies

$${\displaystyle {\mathbf{SCAL}}_{\varphi ,M}^{k}V=\sum _{j=1}^n{\mathbf{S}}_{\varphi ,M}^D\left(v_j\right).}$$

(122)

Proof. 

We have

$$\begin{array}{ccc}{\mathbf{SCAL}}_{\varphi ,M}^{k}V(x,t)\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\sum _{j=1}^n{v}_j\left(t\right)\overline{v_j\left(y\right){\varphi }_{x,t}^M\left(t\right)}{\varphi }_{x,t}^M\left(y\right){\gamma }_k\left(dt\right){\gamma }_k\left(dy\right)}\hfill \\ & =\hfill & {\displaystyle \sum _{j=1}^n{〈v_j,{\varphi }_{x,t}^M〉}_{L_k^2\left({\mathbb{R}}^N\right)}{\overline{〈v_j,{\varphi }_{x,t}^M〉}}_{L_k^2\left({\mathbb{R}}^N\right)}}\hfill \\ & =\hfill & {\displaystyle \sum _{j=1}^n{{\rm Y}}_{\varphi }^M\left(v_j\right)(x,t)\overline{{{\rm Y}}_{\varphi }^M\left(v_j\right)(x,t)}}\hfill \\ & =\hfill & {\displaystyle \sum _{j=1}^n{|{{\rm Y}}_{\varphi }^M\left(v_j\right)(x,t)|}^2.}\hfill \end{array}$$

This allows us to conclude. □

Definition 7. 

We define the time-frequency concentration of a subspace V in R as

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V\right):=\frac{1}{n}{\int }_R{\mathbf{SCAL}}_{\varphi ,M}^{k}V(r,x)d{\mu }_k(r,x).}$$

(123)

Then from Equation (122),

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V\right):=\frac{1}{n}\sum _{i=1}^n{\int }_R{\mathbf{S}}_{\varphi ,M}^D\left(v_i\right)(r,x)d{\mu }_k(r,x).}$$

(124)

Theorem 8. 

The n-dimensional signal space $V_n=\mathrm{span}{\left\{{\psi }_i^R\right\}}_{i=1}^n$ spanned by the first n eigenfunctions of ${\mathcal{L}}_{\varphi }^M\left(R\right)$ corresponding to the n largest eigenvalues ${\left\{{\alpha }_i\left(R\right)\right\}}_{i=1}^n$ maximize the regional concentration ${\zeta }_{R,\varphi }^M\left(V\right)$ and we have

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V_n\right):=\frac{1}{n}\sum _{i=1}^n{\alpha }_i\left(R\right).}$$

(125)

Proof. 

Since

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V_n\right):=\frac{1}{n}\sum _{i=1}^n{\int }_R{\mathbf{S}}_{\varphi ,M}^D\left({\psi }_i^R\right)(r,x)d{\mu }_k(r,x),}$$

(126)

then, according to the min-max lemma for self-adjoint operators (see, for example, Sec.95 in Ref. [31]), we have

$$\begin{array}{ccc}\hfill {\alpha }_i\left(R\right)& =& {\displaystyle {\int }_R{\mathbf{S}}_{\varphi ,M}^D\left({\psi }_i^R\right)\left(z\right)d{\mu }_k\left(z\right)}\hfill \\ & =& max\left\{{〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left(g\right),g〉}_{L_k^2\left({\mathbb{R}}^N\right)}:{\parallel g\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1,g\perp {\psi }_1^R,\dots ,{\psi }_{i-1}^R\right\}.\hfill \end{array}$$

So the eigenvalues of ${\mathcal{L}}_{\varphi }^M\left(R\right)$ determines the number of orthogonal functions that have a well-concentrated scalogram in R. Thus

$${\displaystyle {\zeta }_{R,\varphi }^M\left(V_n\right)=\frac{1}{n}\sum _{j=1}^n{\alpha }_j\left(R\right).}$$

(127)

The min-max characterization of the eigenvalues of compact operators implies that the first n eigenfunctions of the time-frequency operator ${\mathcal{L}}_{\varphi }^M\left(R\right)$ have optimal cumulative time-frequency concentration inside R, in the sense that

$${\displaystyle \sum _{i=1}^n{〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left({\psi }_i^R\right),{\psi }_i^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}=max\left\{{\displaystyle \sum _{i=1}^n{〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left({\kappa }_i\right),{\kappa }_i〉}_{L_k^2\left({\mathbb{R}}^N\right)}:{\left\{{\kappa }_i\right\}}_{i=1}^n\mathrm{orthonormal}}\right\}.}$$

Hence, $V_n$ is the better concentrated in R than any n-dimensional subset V of $L_k^2\left({\mathbb{R}}^N\right)$, i.e.,

$${\zeta }_{R,\varphi }^M\left(V\right)\le {\zeta }_{R,\varphi }^M\left(V_n\right).$$

(128)

The proof is complete. □

Remark 2. 

The time-frequency concentration of the subspace $V_n$ in R satisfies

$${\alpha }_n\left(R\right)\le {\zeta }_{R,\varphi }^M\left(V_n\right)\le {\alpha }_1\left(R\right)\le 1.$$

(129)

3.4. Accumulated Scalogram

Given $n_k(\varphi ,R)=\lceil {\mathfrak{R}}_k(\varphi ,R,M)\rceil$ and $V_{n_k(\varphi ,R)}=\mathrm{span}{\left\{{\psi }_j^R\right\}}_{j=1}^{n_k(\varphi ,R)},$ then the function ${\rho }_{(\varphi ,R,M)}^k:={\mathbf{SCAL}}_{\varphi ,M}^k{V}_{n_k(\varphi ,R)}$ is called the accumulated scalogram, which satisfies

$${\displaystyle {\rho }_{(\varphi ,R,M)}^k=\sum _{j=1}^{n_k(\varphi ,R)}{\left|{{\rm Y}}_{\varphi }^M\left({\psi }_j^R\right)\right|}^2.}$$

(130)

Note that

$${∥{\rho }_{(\varphi ,R,M)}^k∥}_{L_{{\mu }_k}^1(\Omega )}=c(b,k,\varphi )n_k(\varphi ,R)=c(b,k,\varphi ){\mathfrak{R}}_k(\varphi ,R,M)+O\left(1\right).$$

Moreover, since

$${\displaystyle \sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)\le tr\left({\mathcal{L}}_{\varphi }^M\left(R\right)\right)={\mathfrak{R}}_k(\varphi ,R,M)}$$

then

$$F_k^M(\varphi ,R):=1-{\displaystyle \frac{{\displaystyle \sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)}}{{\mathfrak{R}}_k(\varphi ,R,M)}}.$$

(131)

satisfies

$$0\le F_k^M(\varphi ,R)\le 1.$$

(132)

More precisely, we have this result.

Lemma 2. 

Let $\epsilon \in (0,1)$. We have

$$0\le F_k^M(\varphi ,R)\le 1-(1-\epsilon )min\left(1,{\displaystyle \frac{d(\epsilon ,R)}{{\mathfrak{R}}_k(\varphi ,R,M)}}\right).$$

(133)

Proof. 

Let ${\ell }_k(\epsilon ,R)=min(d(\epsilon ,R),n_k(\varphi ,R))$. Then for all $1\le j\le {\ell }_k(\epsilon ,R)$,

$${\alpha }_j\left(R\right)\ge 1-\epsilon ,$$

(134)

and

$${\displaystyle \sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)\ge \sum _{j=1}^{{\ell }_k(\epsilon ,R)}{\alpha }_j\left(R\right)\ge (1-\epsilon ){\ell }_k(\epsilon ,R).}$$

(135)

Thus

$$F_k^M(\varphi ,R)\le 1+(\epsilon -1){\ell }_k(\epsilon ,R){\mathfrak{R}}_k{(\varphi ,R,M)}^{-1}.$$

(136)

As $n_k(\varphi ,R)\ge {\mathfrak{R}}_k(\varphi ,R,M),$ the proof is complete. □

Therefore, when the eigenvalues ${\left\{{\alpha }_j\left(R\right)\right\}}_{j=0}^{d(\epsilon ,R)}\to 1$, then $F_k^M(\varphi ,R)\to 0.$ Additionally, the error between ${\Theta }_k^M$ and ${\rho }_{(\varphi ,R,M)}^k$ is restricted by the result.

Theorem 9. 

We have

$${\displaystyle \frac{1}{{\mathfrak{R}}_k(\varphi ,R,M)}}{∥{\rho }_{(\varphi ,R,M)}^k-{\Theta }_M^k∥}_{L_{{\mu }_k}^1(\Omega )}\le c(b,k,\varphi )\left({\displaystyle \frac{1}{{\mathfrak{R}}_k(\varphi ,R,M)}}+2F_k^M(\varphi ,R)\right).$$

(137)

Proof. 

From Lemma 1,

$${\displaystyle {\rho }_{(\varphi ,R)}^k\left(z\right)-{\Theta }_M^k\left(z\right)=\sum _{j=1}^{\infty }({\beta }_j-{\alpha }_j\left(R\right)){|{{\rm Y}}_{\varphi }^M\left({\psi }_j^R\right)\left(z\right)|}^2,z=(x,t)\in R,}$$

(138)

in which ${\beta }_j=1$ for $j\le n_k(\varphi ,R)$ and ${\beta }_j=0$ for $j>n_k(\varphi ,R)$. Now since

$${\displaystyle {∥|{{\rm Y}}_{\varphi }^M\left({\psi }_j^R\right)(x,t){|}^2∥}_{L_{{\mu }_k}^1(\Omega )}=c(b,k,\varphi )\mathrm{and}\sum _{j=1}^{\infty }{\alpha }_j\left(R\right)={\mathfrak{R}}_k(\varphi ,R,M),}$$

then

$$\begin{array}{ccc}& & {∥{\rho }_{(\varphi ,R)}^k-{\Theta }_M^k∥}_{L_{{\mu }_k}^1(\Omega )}\hfill \\ & \le & {\displaystyle c(b,k,\varphi )\sum _{j=1}^{\infty }|{\beta }_j-{\alpha }_j\left(R\right)|}\hfill \\ & =& c(b,k,\varphi )\left({\displaystyle \sum _{j=1}^{n_k(\varphi ,R)}(1-{\alpha }_j\left(R\right))+\sum _{j>n_k(\varphi ,R)}{\alpha }_j\left(R\right)}\right)\hfill \\ & =& c(b,k,\varphi )\left({\displaystyle n_k(\varphi ,R)+\sum _{j=1}^{\infty }{\alpha }_j\left(R\right)-2\sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)}\right)\hfill \\ & =& c(b,k,\varphi )\left(\left(n_k(\varphi ,R)-{\mathfrak{R}}_k(\varphi ,R,M)\right)+2\left({\displaystyle {\mathfrak{R}}_k(\varphi ,R,M)-\sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)}\right)\right)\hfill \\ & \le & c(b,k,\varphi )\left(1+2\left({\displaystyle {\mathfrak{R}}_k(\varphi ,R,M)-\sum _{j=1}^{n_k(\varphi ,R)}{\alpha }_j\left(R\right)}\right)\right),\hfill \end{array}$$

and the estimate (137) follows. □

3.5. Approximation Relations

If $V_n=\mathrm{span}{\left\{{\psi }_j^R\right\}}_{j=1}^n$, then for $f\in V_n$ we have

$$\begin{array}{ccc}\hfill {\langle {\mathcal{L}}_{\varphi }^M\left(R\right)f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}& =& \sum _{j=1}^n{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\hfill \\ & \ge & {\alpha }_n\left(R\right)\sum _{j=1}^n{\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\hfill \\ & =& {\alpha }_n\left(R\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

(139)

This means that $f\in {\mathcal{C}}_{1-{\alpha }_n\left(R\right),\varphi }\left(R\right)$. Thus

$$V_n\subset {\mathcal{C}}_{1-{\alpha }_n\left(R\right),\varphi }\left(R\right).$$

(140)

On the other hand, we have

$${\mathcal{V}}_{d(\epsilon ,R)}\subset {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right).$$

(141)

Moreover, since

$${〈{\mathcal{L}}_{\varphi }^M\left(R\right)\left({\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_n^R\right),{\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_j^R〉}_{L_k^2\left({\mathbb{R}}^N\right)}={\alpha }_j{\left(R\right)}^3={\alpha }_j\left(R\right){∥{\mathcal{L}}_{\varphi }^M\left(R\right){\psi }_j^R∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(142)

then by Proposition 12, the function ${\mathcal{L}}_{\varphi }^M\left(R\right)\left({\psi }_j^R\right)$ is in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, if and only if $j\le d(\epsilon ,R)$. Thus if $j\ge 1+d(\epsilon ,R)$, then neither ${\psi }_j^R$ nor ${\mathcal{L}}_{\varphi }^M\left(R\right)\left({\psi }_j^R\right)$ are in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$.

In contrast, functions in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ are not necessarily in ${\mathcal{V}}_{d(\epsilon ,R)}$. Nevertheless the result below characterizes functions in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$.

Theorem 10. 

For $f\in L_k^2\left({\mathbb{R}}^N\right)$, we denote by ${\mathcal{P}}_{\mathrm{Ker}}\left(f\right)$ the orthogonal projection of f onto the kernel $\mathrm{Ker}\left({\mathcal{L}}_{\varphi }^M\left(R\right)\right)$ of ${\mathcal{L}}_{\varphi }^M\left(R\right)$. Then $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ if and only if, it satisfies

$$\begin{array}{cc}\hfill \sum _{j=1}^{d(\epsilon ,R)}({\alpha }_j\left(R\right)-1+\epsilon ){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2& \ge (1-\epsilon ){∥{\mathcal{P}}_{\mathrm{Ker}}\left(f\right)∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ \hfill & +\sum _{j=1+d(\epsilon ,R)}^{\infty }(1-\epsilon -{\alpha }_j\left(R\right)){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2.\hfill \end{array}$$

(143)

Proof. 

The eigenfunctions $\{{\psi }_j^R,j\in \mathbb{N}\}$ form an orthonormal subset in $L_k^2\left({\mathbb{R}}^N\right)$ possibly incomplete if $\mathrm{Ker}\left({\mathcal{L}}_{\varphi }^M\left(R\right)\right)\ne \left\{0\right\}$. Thus

$$f=\sum _{j=1}^{\infty }{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}{\psi }_j^R+{\mathcal{P}}_{\mathrm{Ker}}\left(f\right).$$

(144)

Since ${\mathcal{L}}_{\varphi }^M\left(R\right)\left({\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\right)=0$, then

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right),f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}=\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2.$$

(145)

Hence, by Proposition 12, the function f is in ${\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ if, and only if,

$${\langle {\mathcal{L}}_{\varphi }^M\left(R\right)\left(f\right),f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}=\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\ge (1-\epsilon ){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(146)

As

$${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2={\langle f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+\sum _{j=1}^{\infty }{\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2,$$

then $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ if, and only if,

$$\sum _{j=1}^{\infty }{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\ge (1-\epsilon )\left(\parallel {\mathcal{P}}_{\mathrm{Ker}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+\sum _{j=1}^{\infty }{\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\right),$$

(147)

which allows us to conclude, since the ${\alpha }_j\left(R\right)$’s verify

$$1\ge {\alpha }_1\left(R\right)\ge \cdots \ge {\alpha }_{d(\epsilon ,R)}\left(R\right)\ge 1-\epsilon >{\alpha }_{1+d(\epsilon ,R)}\left(R\right)\ge \cdots >0.$$

(148)

The proof is complete. □

In particular, a function $f\in L_k^2\left({\mathbb{R}}^N\right)$ is in ${\mathcal{C}}_{1-{\alpha }_n\left(R\right),\varphi }\left(R\right)$, if, and only if,

$$\begin{array}{ccc}\hfill \sum _{j=1}^{n-1}({\alpha }_j\left(R\right)-{\alpha }_n\left(R\right)){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2& \ge & {\alpha }_n\left(R\right){∥{\mathcal{P}}_{\mathrm{Ker}}\left(f\right)∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & +& \sum _{j=1}^{n+1}\left({\alpha }_n\left(R\right)-{\alpha }_j\left(R\right)\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2.\hfill \end{array}$$

While a function $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ does not necessarily lies in some subspace $V_n$ of eigenfunctions, a finite number of these eigenfunctions can be used to approximate it.

Theorem 11. 

Fix $\tilde{\epsilon }\in (0,1)$. If $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$, then,

$${∥f-\sum _{j=1}^{d(\tilde{\epsilon },R)}{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}{\psi }_j^R∥}_{L_k^2\left({\mathbb{R}}^N\right)}\le \sqrt{{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}}\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(149)

Proof. 

Let $\mathcal{P}$ denote the orthogonal projection onto the subspace $V_{d(\tilde{\epsilon },R)}$. We have

$$\mathcal{P}f=\sum _{j=1}^{d(\tilde{\epsilon },R)}{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}{\psi }_j^R$$

and

$${∥f-\mathcal{P}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}^2=\sum _{1+d(\tilde{\epsilon },R)}^{\infty }{\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2={\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

First we will prove that

$${\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\ge \left(1-{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(150)

Assuming by contradiction that

$${\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2=\sum _{j=1}^{d(\tilde{\epsilon },R)}{\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2<\left(1-{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(151)

Then

$$\sum _{1+d(\tilde{\epsilon },R)}^{\infty }{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2<(1-\tilde{\epsilon })\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\right),$$

Therefore,

$$\begin{array}{ccc}\hfill {\langle {\mathcal{L}}_{\varphi }^M\left(R\right)f,f\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}& =& \sum _{j=1}^{\infty }{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\hfill \\ & <& {\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+(1-\tilde{\epsilon })\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\right)\hfill \\ & <& (1-\tilde{\epsilon }){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+\tilde{\epsilon }(1-\epsilon /\tilde{\epsilon }){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-(1-\tilde{\epsilon }){\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & =& {(1-\epsilon )\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-(1-\tilde{\epsilon }){\parallel {\mathcal{P}}_{\mathrm{Ker}}\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & <& {(1-\epsilon )\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

Thus $f\notin {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$ is a contradiction.

Now, since

$${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2={\parallel \mathcal{P}f+(f-\mathcal{P}f)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2={\parallel \mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2+{\parallel f-\mathcal{P}f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,$$

(152)

then we get the intended inequality. □

From the previous theorem we can deduce that, for all $f\in {\mathcal{C}}_{\epsilon ,\varphi }\left(R\right)$,

$${〈{\mathcal{L}}_{\varphi }^M\left(R\right)\mathcal{P}f,\mathcal{P}f〉}_{L_k^2\left({\mathbb{R}}^N\right)}=\sum _{j=1}^{d(\tilde{\epsilon },R)}{\alpha }_j\left(R\right){\left|{\langle f,{\psi }_j^R\rangle }_{L_k^2\left({\mathbb{R}}^N\right)}\right|}^2\ge \left(1-\tilde{\epsilon }\right)\left(1-{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.$$

(153)

4. Conclusions and Perspectives

Highly concentrated signals (functions in $L_k^2\left({\mathbb{R}}^N\right)$) in a region of interest are an essential tool in signal processing, and have many applications in quantum physics and engineering. However, most signals appearing in practise are essentially concentrated, and cannot be expressed explicitly. Moreover, it is well-known that the uncertainty principles in Fourier analysis sets a limit to the possible simultaneous concentration of a function and its LCDT. That is why, to overcome the lack of concentration in the phase-space plane, we have used the tool of DCLWT and concentration operators, to measure their time-frequency content on some subset of finite measure. In particular, we have introduced and studied the scalogram associated with the DCLWT. We show that the signal space generated by the first eigenfunctions of the concentration operator have the maximum localized scalogram in a region of finite measure in the phase-space plane. Then we show that any essentially concentrated function can be approximated by a linear combination of such eigenfunctions.

Of course, our study can not work for general localization operators. The choice of the concentration operator in the special case of the Hilbert space $L_k^2\left({\mathbb{R}}^N\right)$ is important in our study due to the specific characteristics of this space, and the choice of the special symbol ${\chi }_R$ is related to the region of interest R.

We believe that any tool used is restricted under certain conditions; therefore, in future work, we will study some qualitative or quantitative uncertainty principles for the DCLWT, which set a limitation to the maximal phase-space resolution of some signals.