1. Introduction and Preliminaries
The Gabor theory has gained respectable status in the field of time–frequency analysis in a short period of time, since the Gabor transform is superior to the Fourier transform, because of its ability to measure the time–frequency variations of a signal at different time–frequency resolutions.
However, the uncertainty principles in Fourier analysis set a limit to the maximal time–frequency resolution. To overcome this problem, Daubechies [
1] introduced the concept of localization operators, which have been shown to be effective in localizing signals on the time–frequency plane. These operators are also called Toeplitz operators or short-time Fourier multipliers.
Extensive research on localization operators has been conducted, notably by Slepian and Pollak [
2] in the classical setting, with further contributions by Landau and Pollak [
3,
4], as well as by Slepian [
5,
6]. Many generalizations and extensions can be found in the literature, such as for the Dunkl transform, the Weinstein transform [
7,
8,
9,
10] and the multivariable Bessel transform [
11].
To be more precise, we first recall some results related to the Dunkl theory (see [
12,
13,
14]). Let
be the Euclidean space equipped with a scalar product
and let
. For
in
, let
be the reflection in the hyperplane
orthogonal to
, i.e., for
,
A finite set is called a root system if and for all . For a given root system R, reflections , generate a finite group , called the reflection group associated with R. We fix and define a positive root system . We normalize each as .
A function
is called a multiplicity function if it is invariant under the action of
W, and we introduce the index
as
Throughout this paper, we will assume that
for all
, and we will denote by
the weight function on
given by
which is invariant and homogeneous of degree
.
The Dunkl operators
,
, on
related to the multiplicity function
k and the positive root system
are as follows:
Specifically, we see that the Dunkl operators commute pairwise and are skew-symmetric with respect to the W-invariant measure .
If
, then the Dunkl–Laplace operator
is defined by:
where ∇ and Δ are, respectively, the nabla and Euclidean Laplacian operators.
Like spherical functions on a Riemannian symmetric space, for all
, the following system
has a unique analytic solution
. This solution is known as the Dunkl kernel (also called nonsymmetric generalized Bessel function). Observe, moreover, that the Dunkl kernel has symmetric arguments and is holomorphic on
. This kernel is explicitly known only in very few cases, that include, for example, the rank-one case and the symmetric group
.
The Dunkl–Bessel transform [
15] is defined by
where
and
is the measure on
defined by
Here, is the normalized Bessel function and K is the Dunkl kernel.
This transformation generalizes the Fourier transform, the Weinstein transform and the multivariable Bessel transform, and has been studied in several mathematical problems, such as the mean value theorem [
15], uncertainty principles [
16], wavelet theory [
17] and so on.
As for the Fourier transform, the Dunkl–Bessel transform has shown some weakness in the study of signals on the time–frequency plane, and this fact has been illustrated by the various formulations of the uncertainty principle in this setting.
The aim of this paper is to follow Daubechies’ approach, by studying first the Dunkl–Bessel Gabor transform, and then introducing the Toeplitz operator in this context. We will show some harmonic analysis results for the Dunkl–Bessel Gabor transform, such as a Parseval-type equality and an inversion formula. Then, we will give criteria on the compactness and boundedness of the introduced Toeplitz operators. As a special case of these operators, we will focus on concentration operators, which are compact and self-adjoint and even trace-class. As in the case of the prolate spheroidal wave functions, we will show that their eigenfunctions are maximally time–frequency-concentrated on the region of interest from the time–frequency plane.
Typical examples of higher dimensional signals include image and video signals. Indeed, a video signal is a sequence of individual video frames, or images. Unlike one-dimensional signals, there are special problems that arise in higher dimensional signals, such as edge detection, target detection and tracking, and special techniques are required to deal with such problems. It is therefore worthwhile to delve into the field of higher dimensional time–frequency analysis both from theoretical and applied perspectives. The theoretical considerations in the proposal shall encompass the formulation of novel tools of time–frequency analysis, particularly well suited for higher dimensional signals arising in numerous scientific pursuits. The application areas shall be primarily within the field of signal and image processing, with special emphasis on distinct sectors such as energy and economics.
The structure of this manuscript is as follow: In
Section 2, we define the Dunkl–Bessel Gabor transform and study its harmonic analysis. Then, we introduce and study the Toeplitz operators in
Section 3. Next, we examine the
-boundedness and compactness of these operators, under some appropriate conditions on the symbol and the window function. Finally, in
Section 4, we investigate some spectral analysis issues related to concentration operators.
2. The Dunkl–Bessel Gabor Transform
We will first recall the basic results on the Dunkl–Bessel transform (cf. [
15]). To do this, let
and let
.
If
, then for all
we define:
where
is the space of functions of class
on
, even with respect to the last variable. This operator is known as the Dunkl–Bessel–Laplace operator, where
is the Dunkl–Laplace operator on
and
the Bessel operator on
given by
The Dunkl–Bessel kernel
is given by
where
is the normalized Bessel function and
K is the Dunkl kernel.
This kernel satisfies the following properties:
For all
and for all
, we have
For all
and
, we have
where
and
In particular, for all
,
We denote by
the space of measurable functions on
such that:
where
is the measure on
defined by
and
If
, this space is provided with the following scalar product:
For
, the Dunkl–Bessel transform is defined by:
This transformation satisfies the following properties:
For any
,
For all
such that
belongs to
, we have
where is the Schwartz space of rapidly decreasing functions on , even with respect to the last variable.
Moreover we have the following Plancherel-type and Parseval-type formulas for the Dunkl–Bessel transform.
Proposition 1. - 1.
Plancherel’s formula: The Dunkl–Bessel transform is a topological isomorphism from onto itself and for all f in , In particular, the Dunkl–Bessel transform can be uniquely extended to an isometric isomorphism on .
- 2.
Parseval’s formula: For all , we have
Next, we will introduce the generalized translation and the convolution operator by the use of the Dunkl–Bessel kernel. That is, the Dunkl–Bessel translation
is defined by:
In the following theorem, we recall the main properties of
established on the subspace
of radial functions in
,
(see [
15]).
Theorem 1. - 1.
Let be a nonnegative function. Then, we haveand - 2.
For all , , we have
The Dunkl–Bessel convolution product
of the two functions
is defined by:
This convolution is commutative, associative and satisfies the following properties.
Proposition 2. Let such that
- 1.
For all , belongs to and - 2.
If and is radial, then such that - 3.
If , then the function belongs to if and only if the function belongs to and (19) holds. - 4.
If , thenwhere both members are finite or infinite.
Now, to introduce the Dunkl–Bessel Gabor transform, we will fix some notation: Let
and we will denote by
,
, the space of measurable functions
f on
satisfying
where
.
Definition 1. For , we define the modulation of any function bywhere is the generalized translation operator. Equality (
22) may be easily confirmed by the positivity of the translation operator
on radial functions, as shown by Theorem 1. Furthermore, based on (
13) and (
16), we have for all
For
and
, we define a new family of functions
as follows:
Definition 2. Let . Then, the Dunkl–Bessel Gabor transform , with respect to h, is defined byfor any function . It can also be expressed via the convolution product (
18) as
where for
,
.
One can easily see that for any
and
, we have
where
Theorem 2. Let . Then, for all we have:
- 1.
The function belongs to such that - 2.
A Plancherel-type formula: - 3.
- 4.
Proof. The first statement can easily obtained from the Cauchy–Schwarz inequality and Inequality (
25). Then, by (
21), (
27), (
12), (
22) and Plancherel’s formula (
13), we have
This implies (
31). Finally, Inequality (
32) follows by applying the Riesz–Thorin interpolation theorem and the relations (
29) and (
30). □
Proposition 3. Let . Then, is a reproducing kernel Hilbert space embedded as a subspace of , with kernel It is pointwise bounded, that is: Proof. By Parseval’s formula (
31), we obtain
Then, by Proposition 2, we can easily deduce that for every
, the function
belongs to
. This achieves the proof. □
Now, in order to state an inversion formula for the Dunkl–Bessel Gabor transform, we need to establish a few lemmas.
Lemma 1. For a nonzero function and N a positive integer, we define the functions and by:andwhere Proof. By Cauchy–Schwarz’s inequality, the sequence
satisfies
On the other hand, by Proposition 2 and relations (
7), (
13), (
12) and (
22), we obtain:
This implies that
. Therefore, by (
16) we obtain
which means that
. Finally, for all
,
This achieves the proof. □
Lemma 2. The function can be written as Proof. By Proposition 2, we have
This gives the desired result. □
Lemma 3. Let h be in and N be any positive integer. Then, for all , we havewhere is defined by Proof. Since by Proposition 2 we have
, then by using Young’s inequality and Parseval’s formula, we obtain
and
The proof is complete. □
We will now state the main result of this section.
Theorem 3 (inversion formula).
Let such that . Then, for all , the function belongs to and satisfies Proof. By Proposition 2, Lemma 1 and Equality (
39), we derive that
and
On the other hand, by (
13) we obtain that
pointwise as
. Thus,
which gives the desired result. □
3. Toeplitz Operators for the Dunkl-Bessel Gabor Transform
We begin this section by recalling some notions on Schatten–von Neumann classes. We will denote by
the set of all infinite sequences of real (or complex) numbers
, such that
For
, we provide the space
with the scalar product
If we denote by
the space of bounded operators from
into itself, then the singular values
of the compact operator
are the eigenvalues of the positive self-adjoint operator
. In particular, for
, the Schatten class
is the space of all compact operators whose singular values lie in
. This space is equipped with the norm
Notice that
is the space of trace-class operators, and
is the space of Hilbert–Schmidt operators. Thus, if
T is an operator in
, and
is an orthonormal basis of
, then the trace of
T is defined by:
In particular, if
T is positive, then its trace satisfies
Furthermore, if the positive operator
is in
, then any compact operator
T on
is Hilbert–Schmidt. Following that, for any orthonormal basis
of
, we have
We set
, the space of bounded operators equipped with the standard norm:
Now, in order the prove the main results of this section, we will introduce the Toeplitz (or time–frequency) operator into our setting. To do this, let
be a measurable function on
and
h be a radial measurable function on
; then, the Toeplitz operator
related to the Dunkl–Bessel Gabor transform on
,
is defined by
The last relation is often better understood in a weak sense, that is, for any
and
,
,
For
, the adjoint of the operator
is
such that
In the remaining part of this section, h is a function in such that
Proposition 4. If , then is in with Proof. From (
47) and (
29), we have for all
,
Then, by (
45), we obtain the desired result. □
Proposition 5. If , then the Toeplitz operator is in and we have Proof. For
,
Thus, by (
45) we obtain the desired result. □
Theorem 4. Let and let . Then, there is a unique bounded linear operator such that Proof. For
, we consider the operator
defined by
Then, by Propositions 4 and 5, we obtain
and
Thus, by the Riesz–Thorin interpolation theorem (see [
18] [Theorem 2] and [
19] [Theorem 2.11]),
Since (
52) is true for arbitrary function
f in
, then we obtain the desired result. □
The second aim of this section is to prove that the Toeplitz operator
is in the Schatten class
.
Proposition 6. If , then is a Hilbert–Schmidt operator such that Proof. Assume that
is an orthonormal basis for
. Next, by using Parseval’s identity and relations (
26), (
47) and (
48), we obtain
Therefore, from (
48), (
49) and (
25), we derive
Hence, according to (
53) and [
19] [Proposition 2.8], the operator
belongs to
and is then compact. □
Proposition 7. Let and let Then, the operator is compact.
Proof. Let
be a sequence in
that converges to
in
. Then, Theorem 4 implies
Therefore, in , we have as . Thus, for all j, the operator is compact by Proposition 6. Given that compact operators form a closed subspace of , it follows that is also compact. □
Theorem 5. Let . Then,
- 1.
The operator belongs to andwhere is given by - 2.
This trace formula occurs:
Proof. If
, then, according to Proposition 6, we have
. Moreover, by [
19] [Theorem 2.2]), the kernel’s orthogonal complement of
has an orthonormal basis
, which is made up of eigenvectors of
and
an orthonormal set in
, such that
where the positive singular values of
corresponding to
are
. Next, we obtain
Therefore, using Bessel and Cauchy–Schwarz’s inequalities together with relations (
25) and (
26), we obtain
Thus,
On the other hand, we have that
belongs to
, and by (
56), we have
Then, using Fubini’s theorem
This allows us to conclude.
Now, assume that
is an orthonormal basis for
. Then, the operator
. Therefore, using (
42) and Parseval’s identity,
This achieves the proof. □
Involving Theorem 5 and Proposition 5 and using an interpolation argument (see [
19] [Theorems 2.10 and 2.11]), we deduce the following result.
Corollary 1. If , then the Toeplitz operator belongs to and Remark 1. If is positive and real-valued, then is a positive operator. Moreover, using (43) and (55), we obtain Corollary 2. If are positive and real-valued, then the trace-class operators are positive, such that for all : Proof. If
are positive, then (see [
20] [Theorem 1]),
Let and . Then, by Remark 1, we obtain the desired result. □
4. Spectral Analysis for the Generalized Concentration Operator
In this section,
h will be any function of
such that
. We define
the adjoint of
by
We denote by
the orthogonal projection from
onto
and
the orthogonal projection from
onto the subspace of functions supported in the subset
, that is,
where
denotes the characteristic function of
U.
We define the concentration operator
by
where
, and
U is a subset of
with finite measure
.
We have
, and by (
33), the operator
can be written as follows
On the other hand, since
is the integral kernel of an orthogonal projection, then we have
. Therefore,
Moreover, for any orthonormal basis
of
, the kernel
can be represented as follows
4.1. Dunkl–Bessel Gabor Toeplitz Operator
Definition 3. If is the Dunkl–Bessel Gabor transform of the function with respect to the window function , then the spectrogram of f is defined by From the Plancherel-type formula for
, we have
Spectrogram (
60) is interpreted as a time–frequency energy density of the function
, and by (
47), we have
Definition 4. We define the Dunkl–Bessel Gabor Toeplitz operatorby Proposition 8. The operator is bounded and positive, satisfyingand Proof. For any
F in
, we have
Then, we obtain (
64). On the other hand, by (
64), it follows that
is positive and bounded.
Since the operator
can be written as
then we obtain
The proof is complete. □
Proposition 9. The operator is compact and trace-class, satisfyingwhere Proof. We know that
is a positive bounded operator. If
is any orthonormal basis for
, then
is an orthonormal basis for
, where
. Thus, by (
47) and the Fubini’s theorem, we obtain
Hence, by (
41), (
42) and (
44), the operator
is trace-class such that
This gives the desired result. □
Let be the operator defined by . This operator is more convenient than the operator , since its domain is the whole space , and then its spectral properties can be obtained from its integral kernel.
As
is positive and trace-class, then by the following decomposition:
it follows that
is also positive and trace-class, satisfying
Proposition 10. The trace of is given by the following relation Proof. Since
is positive, then
Moreover, as
is a reproducing kernel Hilbert space with kernel
, then for
,
Therefore,
has the integral kernel:
This completes the proof. □
4.2. Eigenvalues and Eigenfunctions
Since the concentration operator
is compact and self-adjoint, then the spectral theorem gives the following spectral representation:
where
denotes the positive eigenvalues arranged in a non-increasing way and
its corresponding orthonormal set of eigenfunctions. As
, then by (
49) we obtain
Then, by (
65), we obtain that the operator
can be diagonalized as
where
.
Lemma 4. For all we have Proof. By Proposition 3, we have that for any
, the function
belongs to
. Therefore,
If
is an orthonormal basis for
. Then,
is an orthonormal basis of
, and then the reproducing kernel
can be written as follows
The proof is complete. □
Proposition 11. Then, the eigenvalue distribution has the following estimate: Proof. The proof is obtained directly by adaptation of the proof of Lemma 3.3 in [
21]. □
4.3. Spectrogram of a Subspace
Given
V as a
N-dimensional subspace of
, then the orthogonal projection
onto
V, with projection kernel
is given by:
where the kernel
is defined by
for any orthonormal basis
of
V.
Definition 5. The spectrogram of the subspace V of , with respect to the window function h, is denoted by , and is defined by Lemma 5. If V is a subspace of , then the spectrogram can be written as Proof. That achieves the proof. □
Let
be the time–frequency regional concentration of the subspace
V in
U. Then, it is defined by
By Lemma 5, it can be written as follows
Theorem 6. The regional concentration is maximized by the N-dimensional signal space , which is composed of the first N eigenfunctions of corresponding to the N greatest eigenvalues , and we have Proof. Using the min-max lemma for self-adjoint operators (see [
22] [Section 95]), we obtain
Therefore, the number of orthogonal functions with a well-concentrated spectrogram in
U is determined by the eigenvalues of
. Then, we obtain
The first
N eigenfunctions of the time–frequency operator
have the best cumulative time–frequency concentration inside
U, according to the min-max characterisation of the eigenvalues of compact operators. That is,
Consequently,
can be more concentrated in
U than any
N-dimensional subset
V of
; that is,
The proof is complete. □
Remark 2. The time–frequency concentration of a subspace in U satisfies 4.4. Accumulated Spectrogram
We define the accumulated spectrogram by
where
is the smallest integer greater than or equal to
and
Consequently, we obtain the following result.
Lemma 6. If , then we have Proof. For
, we define
. Then, it follows that
Now, since
, then
Hence, since , we derive the desired result. □
Notice that when the eigenvalues are around 1. Furthermore, the error bounding the difference between and is given as follows.
Proposition 12. The following error estimate holds: Proof. For any
, Lemma 4 implies
where
is given by
Since
then
that achieves the proof. □
5. Conclusions and Perspectives
In the present paper, we studied the harmonic analysis related to the Dunkl–Bessel Gabor transform and proved in particular a Plancherel-type formula and an inversion formula. Next, we introduced the Toeplitz operator in the Dunkl–Bessel Gabor setting, and studied its boundedness, compactness and Schatten class properties. Finally, we studied the spectral analysis of the generalized concentration operator, which is compact and self-adjoint. This allowed us to study the optimization problem, which aims to look for functions that have a well-concentrated spectrogram in a subset of the time–frequency plane. Then, we showed that the N-dimensional signal space, consisting of the first N eigenfunctions of the generalized concentration operator corresponding to the N largest eigenvalues, maximizes the concentration of the region of interest.
In this paper, we showed that there is a limitation of maximal time–frequency resolution. As a perspective, one can prove some qualitative or quantitative uncertainty principles for the Dunkl–Bessel Gabor transform, that set restrictions to the possible concentration of a function on the time–frequency plane.