1. Introduction
Throughout this paper we will work on
with a Lebesgue measure
. We denote
as the Schwartz class of functions on
. For
,
denotes the usual Lebesgue space. Let
be Banach algebra, i.e., complete normed space, algebra, and
for all
. A Banach space
is called Banach module over
X if
B is a module over
X in the algebraic sense for some multiplication,
, and satisfies
for all
,
. The distribution function of
f is defined by:
The rearrangement function of
f is defined by:
The average function of
f is also given by
for
. The Lorentz space
is defined as the set of all (equivalence classes) measurable functions
f on
such that
where:
and
The Lorentz space is normed vector space with the norm defined by:
and
in [
1,
2]. In addition, the Lorentz space
is a Banach module over
[
3].
Let us give the definition of the fractional Fourier transform, which is a generalization of the classical Fourier transform and is often used in the fields of quantum physics, signal, and image processing. The fractional Fourier transform with angle
of
is defined by:
where the kernel is:
and
The inversion formula of the fractional Fourier transform is also given by:
where
and
The fractional Fourier transform with angle
corresponds to the classical Fourier transform. Indeed the inversion of fractional Fourier transform with angle
is the fractional Fourier transform with angle
[
4].
In wavelet theory,
a is called the scaling parameter, which measures the degree of compression or scale and
b is a translation parameter, which determines the time location of the wavelet function. The mother wavelet function is defined by:
where
denotes the translation operator and
denotes the dilation of
with
a scale [
5]. The mother wavelet takes its name from two important properties of wavelet theory. The term wavelet means a small wave. This implies that it is a window function of a finite length and oscillatory. The term mother means that functions with different supports are derived from a main function. The mother wavelet is defined as a prototype for generating the window functions, that is, zero-valued outside of a chosen set centered at the orjin, symmetric functions [
6]. In addition the fractional mother wavelet function with angle
is given by:
where
,
. The fractional mother wavelet is a generalization of the mother wavelet. Given any
(called wavelet function), the fractional wavelet transform with angle
of
with respect to
is defined by:
for all
. Since the fractional wavelet transform provides the time-frequency information of a signal in the successful way, it is one of the important time-frequency operators. The fractional wavelet transform with angle
corresponds to the classical wavelet transform. The signal
f is reconstructed from its fractional wavelet transform by:
where
(called admissibility condition). The fractional convolution of
f,
is given by:
For
the fractional convolution is equal to classical convolution. The fractional wavelet transform can also be written as convolution:
for all
[
7].
The signal analysis capability of the wavelet transform is limited in the time-frequency plane. The wavelet transform is inefficient to process signals that are not well concentrated in the frequency domain, such as chirp-like signals ubiquitous in nature and man-made systems [
8]. Therefore, a series of new signal processing tools such as fractional Fourier transform and fractional wavelet transform have been proposed to analyze such signals. Although the fractional Fourier transform overcomes this limitation and can provide signal representations in the fractional field, it fails to obtain the local structures of the signal. The fractional wavelet transform finds a solution to the limitations encountered in these transforms. It covers the advantages of the wavelet transform and the fractional Fourier transform. It has the option to provide multi-analysis and represent signals in the fractional area. This allows the fractional wavelet transform to be applied in many signal processing areas such as speech, vision, communication, and radar [
9]. Thus, the fractional wavelet transform is potentially useful with signaling [
10].
In recent years bilinear and multilinear time-frequency operators have been derived. Furthermore, boundedness and convergence of time-frequency operators in spaces such as Lebesgue, Lorentz, weighted variable exponent amalgam spaces, and modulation spaces have been studied [
11,
12,
13,
14]. However, the boundedness of fractional wavelet transform has also been investigated in various function spaces in the literature [
15,
16,
17].
In wavelet theory, since the dilation operator changes the scale, the continuous wavelet transform gives us local information of a signal at any neighborhood of
x time in
a size. The dilation operator preserves the shape of the signal while doing this process. If the scale
a is close enough to zero, this transform acts like a microscope [
5,
18]. For this reason, the continuous wavelet transform is a useful tool in the plane
. Based on this, in this work we introduce multilinear fractional wavelet transform on
using Schwartz functions. We then give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. In the third section and the fourth section, we consider the boundedness of the multilinear fractional wavelet transform on Lebesque and Lorentz spaces under some conditions.
2. Multilinear Fractional Fourier Transform and Multilinear Fractional Wavelet Transform
In this section we will introduce multilinear fractional wavelet transform and multilinear fractional Fourier transform. We then will consider some of their properties.
Let be a multi-wavelet. That means is a wavelet function for . Now, we can give definitions of the multilinear mother wavelet and the multilinear fractional mother wavelet.
Definition 1. Let be a multi-scaling parameter and let be a multi-translation parameter, i.e., scaling parameter and translation parameter for . The multilinear mother wavelet is defined by:for all . The multilinear mother wavelet is derived by the n-fold product of dilation and translation of a wavelet. Throughout the paper, we will take the parameters and as the multi-scaling parameter and multi-translation parameter, respectively. The multilinear fractional mother wavelet with multi-angle is defined as:where , , and . We shall also use the notation In this work f denotes both the vector and the tensor product such that , . We write for the inner product on and use the notation Furthermore, we say that if there exists such that . We will use similar notations throughout this work.
Definition 2. Let and . For all , the multilinear fractional convolution of and is defined as:where , denotes the fractional convolution. Definition 3. Let and . Assume that is a multilinear fractional mother wavelet given by (2). The multilinear fractional wavelet transform of with a multi-angle θ is defined by:for all . Definition 4. Let be a multi-wavelet. If there exists the condition:then it is said that Ψ has a multi-fractional admissiblity condition. In this paper, we will assume that multi-wavelet
has multi-fractional admissiblity condition and
is a multilinear fractional mother wavelet given by (
2).
Theorem 1. Let . Then,andholds for all Proof. Let
be given. For all
, we have:
In adddition, by [
7], it is known that:
for all
,
. Combining (
3) and (
4), we have:
In addition, by (
5), we obtain:
Since
is dense in
, which means closure of
equals to
[
19], we achieve the equalities (
6) and (
7) for all
□
Definition 5. Let . The multilinear fractional Fourier transform of with a multi-angle θ is defined as:where the is kernel given by (1). Sometimes we will denote the multilinear fractional Fourier transform of with a multi-angle θ with the symbol . Since the classical fractional Fourier transform is injective and surjective from to (see in [20]), the multilinear fractional Fourier transform is injective and surjective from to . Now we can give definition of the multilinear inverse fractional Fourier transform of with a multi-angle θ: In addition, the multilinear inverse fractional Fourier transform of with a multi-angle θ is shown with the symbol . Furthermore, since and by Theorem 3.1 in [20], it is easily shown that and . Theorem 2. (Hausdorff-Young inequality for multilinear fractional Fourier transform)
Assume that and . If , then and: Proof. Take any
. It is clear that:
for
. By Parseval’s relation in [
7], we also know:
for
. Then, using (
8), we have:
where
. Now, we will use (
9), then:
So the estimates (
10) and (
11) say that the fractional Fourier transform
is bounded from
into
and from
into
with operator norms
,
respectively. If we take
, then we have
,
,
. So we find
. Now if we use the interpolation theorem for multilinear operators in [
21], we obtain that the fractional Fourier transform
is bounded from
into
with the norm:
On the other hand, we have:
Therefore by (
12) and (
13), we achieve:
□
Theorem 3. (Paley-type inequality for linear fractional Fourier transform)
Assume that , and . If , then and: Proof. It is known that
and
into
are bounded with the operator norms
and
1, respectively. Then, by real interpolation [
21], we achieve:
where
,
,
with the norm:
Hence, we obtain that
is bounded, where
,
. Moreover combining (
13) and (
14), we conclude:
Therefore, the proof is completed. □
Theorem 4. (Paley-type inequality for multilinear fractional Fourier transform)
Assume that , , , and . If , then and Proof. Take any
. Then, we have:
So we can say that the multilinear fractional Fourier transform is a tensor product operator. Furthermore, we can write:
and
for all
. Hence, we find that the multilinear fractional Fourier transform is of restricted weak types (1,...,1;1) and (
∞,...
∞;
∞). If we can generalize Theorem 7.7 in [
21] for multilinear fractional Fourier transform, then there exists
such that:
where
,
and
,
. Using Theorem 3 and by (
15), we also get:
□
Theorem 5. Let , . If , are two multi-wavelets in , then:wherefor Proof. Take any
,
. Assume that
,
are two multi-wavelets in
. It is known that
where
for
[
7]. Then by (
16), we get:
□
Theorem 6. Let , . If is a multi-wavelet in , then: Proof. If we take in Theorem 5, we get the desired result. □
By setting in Theorem 6, the following theorem is obtained.
Theorem 7. Let . If is a multi-wavelet in , then: Let us derive the inversion formula for the multilinear fractional wavelet transform.
Theorem 8. Let . If is a multi-wavelet in , then can be reconstructed by the following equation: Proof. Take any
. We can write:
for
. Then by (
17), we achieve:
□
4. Boundedness of the Multilinear Fractional Wavelet Transform on Lorentz Spaces
The multilinear fractional wavelet transform is a tensor product operator such that:
for all
and
. Take
. It satisfies:
and
for all
. Then multilinear fractional wavelet transform is of restricted weak types (1,...,1;1) and (
∞,...,
∞;
∞). So we can generalize Theorem 7.7 in [
21] for multilinear fractional wavelet transform. Hence, there exists
such that:
where
,
and
,
.
Theorem 13. Let , , and , . Then,
- (i)
The multilinear fractional wavelet transform is bounded from into ;
- (ii)
In addition if , , then the multilinear fractional wavelet transform is bounded from into .
Proof. (i) Take any
. Then by (
28), we have:
On the other hand since
is a Banach module over
[
3] and by (
29), we get:
This is the desired result.
(ii) Let
be given. We then have:
for
. So the rearrangement of
is:
for
. Thus, we have
,
. Therefore, using (
30) and by Theorem 7.6 in [
21], we have:
where
,
. This completes the proof. □
Now, using Theorem 13, we can give the following corollary.
Corollary 3. Let , , and , . Then,
- (i)
The multilinear fractional wavelet transform is bounded from
into ;
- (ii)
In addition if , , then, the multilinear fractional wavelet transform is bounded from into .
Theorem 14. Let , , , and . Assume that . Then, the multilinear fractional wavelet transform is bounded from into for any . Moreover,holds for all Proof. Take any
. Since
, we have
,
. Then, by Theorem 3 and the inequality (
31), we conclude:
where
,
. □
Using the inequalities (
30) and (
31), the following theorems and corollary are proven similar to the proof of Theorems 11 and 12, and Corollary 2.
Theorem 15. Let , , , and let , be multi-wavelets.
(i)Then,holds for all . (ii)In addition if , , then,holds for all Theorem 16. Let , , , and let be multi-wavelets.
(i)Then,holds for all , . (ii)In addition if , , then,holds for all , . Corollary 4. Let , , , and let , be multi-wavelets.
(i)Then,holds for all , (ii)In addition if , , then,holds for all , . 5. Conclusions
This paper was motivated to define the multilinear fractional wavelet transform on cartesian product spaces by taking the multi-convolution of Schwartz functions and their dilations. At the same time, based on the tensor product of the fractional Fourier transforms, the multilinear fractional Fourier transform was derived on cartesian product spaces. Thus many properties of the fractional wavelet transform and the fractional Fourier transform were transferred to a multilinear form. Thanks to these new multilinear fractional transforms, it is possible to examine the signal representations and local structures of signals in the n-dimensional fractional space. These features will give you the chance to make these transforms valuable when working with signals in areas such as speech, vision, communication, signal processing, and radar.
In this paper, using multilinear interpolation techniques, we proved that the fractional wavelet transform defined on the cartesian product of Lebesgue spaces is bounded under some conditions for any multi-scaling parameter. We then investigated the boundedness of this transform on cartesian products of Lorentz spaces by using the property of being a tensor product operator. On the other hand, it is known that the Hausdorff–Young and Paley inequalities are fundamental results about the mapping properties of the Fourier transform on Lebesgue and Lorentz spaces. For this reason, we also proved the Hausdorff–Young inequality and Paley-type inequality for multilinear fractional Fourier transform, again using the multilinear interpolation technique. In our paper, the Hausdorff–Young inequality says us that the multilinear fractional Fourier transform maps continuously into , . The Paley type inequality also tells us that the multilinear fractional Fourier transform maps continuously into , , , , . That means the multilinear fractional Fourier transform is extended to Lebesgue and Lorentz spaces. Thus, these new inequalities become valuable and fundamental results for Fourier analysis.