1. Introduction
Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. One of the most useful methods to construct wavelet frames is through the concept of the unitary extension principle (UEP) introduced by Ron and Shen [
1] and was subsequently extended by Daubechies
et al. [
2] in the form of the oblique extension principle (OEP). They give sufficient conditions for constructing tight wavelet frames for any refinable function
that generates a multiresolution analysis. The resulting wavelet frames are based on multiresolution analysis, and the generators are often called framelets. These methods of construction of wavelet frames are generalized from one dimension to higher dimensions, tight frames to dual frames, from a single scaling function to a scaling function vector. More importantly, the setup of tight wavelet frames provides great flexibility in approximating and representing periodic functions. Using periodization techniques, Zhang [
3] constructed a dual pair of periodic wavelet frames for
under the assumption that the support of the wavelet function
ψ in the frequency domain is contained in
. Later on, Zhang and Saito [
4] constructed general periodic wavelet frames using extension principles. More precisely, they proved that under some decay conditions, the periodization of any wavelet frame constructed by the unitary extension principle is a periodic wavelet frame, and the periodization of any pair of dual wavelet frames constructed by the mixed extension principle is a pair of dual periodic wavelet frames. To mention only a few references on wavelet frames, the reader is referred to [
5,
6,
7,
8] and the many references therein.
The past decade has also witnessed a tremendous interest in the problem of constructing compactly-supported orthonormal scaling functions and wavelets with an arbitrary dilation factor
(see Debnath and Shah [
9]). The motivation comes partly from signal processing and numerical applications, where such wavelets are useful in image compression and feature extraction, because of their small support and multifractal structure. Lang [
10] constructed several examples of compactly-supported wavelets for the Cantor dyadic group by following the procedure of Daubechies [
11] via scaling filters, and these wavelets turn out to be certain lacunary Walsh series on the real line. Kozyrev [
12] found a compactly-supported
p-adic wavelet basis for
, which is an analog of the Haar basis. The concept of multiresolution analysis on a positive half-line
was recently introduced by Farkov [
13]. He pointed out a method for constructing compactly-supported orthogonal
p-wavelets related to the Walsh functions and proved necessary and sufficient conditions for scaling filters with
many terms (
to generate a
p-MRAin
. Subsequently, dyadic wavelet frames on the positive half-line
were constructed by Shah and Debnath in [
14] using the machinery of Walsh–Fourier transforms. They have established necessary and sufficient conditions for the system
to be a frame for
. Wavelet packets and wavelet frame packets related to the Walsh polynomials were deeply investigated by Shah and Debnath in [
14,
15]. Recent results in this direction can also be found in [
16,
17] and the references therein.
Recently, Shah [
18] established a unitary extension principle for constructing normalized tight wavelet frames generated by the Walsh polynomials on
. Drawing inspiration from these wavelet frames, our aim is to extend the notion of wavelet frames to periodic wavelet frames on
by using extension principles. More precisely, we prove that under some mild conditions, the periodization of any wavelet frame constructed by the unitary extension principle is a periodic wavelet frame on a positive half-line
. Furthermore, based on the mixed extension principle and Walsh–Fourier transforms of the wavelet frames, an explicitly-constructed method for a pair of dual periodic wavelet frames generated by the Walsh polynomials is also given.
This paper is organized as follows. In
Section 2, we introduce some notations and preliminaries related to the operations on positive half-line
, including the definitions of the Walsh–Fourier transform and MRA-based wavelet frames related to the Walsh polynomials.
Section 3 and
Section 4 state and prove our main results about periodic wavelet frames generated by the Walsh polynomials.
2. Walsh–Fourier Analysis and MRA-Based Wavelet Frames
We start this section with certain results on Walsh–Fourier analysis. We present a brief review of generalized Walsh functions, Walsh–Fourier transforms and their various properties.
As usual, let
and
. Denote by
the integer part of
x. Let
p be a fixed natural number greater than one. For
and any positive integer
j, we set:
where
. Clearly,
and
are the digits in the
p-expansion of
x:
Moreover, the first sum on the right is always finite. Besides,
where
and
are, respectively, the integral and fractional parts of
x.
Consider on
the addition defined as follows:
with
, where
and
are calculated by Equation (1). Clearly,
and
. As usual, we write
if
, where ⊖ denotes subtraction modulo
p in
.
Let
; we define a function
on
by:
The extension of the function
to
is given by the equality
. Then, the system of generalized Walsh functions
on
is defined by:
where
They have many properties similar to those of the Haar functions and trigonometric series and form a complete orthogonal system. Further, by a Walsh polynomial, we shall mean a finite linear combination of Walsh functions.
For
, let:
where
are given by Equation (1).
We observe that:
and:
where
and
is
p-adic irrational. It is well known that systems
and
are orthonormal bases in
[0,1] (see Golubov
et al. [
19]).
The Walsh–Fourier transform of a function
is defined by:
where
is given by Equation
. The Walsh–Fourier operator
,
, extends uniquely to the whole space
. The properties of the Walsh–Fourier transform are quite similar to those of the classic Fourier transform (see [
19,
20]). In particular, if
, then
and:
Moreover, if
, then we can define the Walsh–Fourier coefficients of
f as:
The series
is called the Walsh–Fourier series of
f. Therefore, from the standard
-theory, we conclude that the Walsh–Fourier series of
f converges to
f in
, and Parseval’s identity holds:
By
p-adic interval
of range
n, we mean intervals of the form:
The
p-adic topology is generated by the collection of
p-adic intervals, and each
p-adic interval is both open and closed under the
p-adic topology (see [
19]). The family
forms a fundamental system of the
p-adic topology on
. Therefore, the generalized Walsh functions
, assume constant values on each
p-adic interval
and, hence, continuous on these intervals. Thus,
for
.
Let
be the space of
p-adic entire functions of order
n, that is the set of all functions that are constant on all
p-adic intervals of range
n. Thus, for every
, we have:
Clearly, each Walsh function of order up to belongs to . The set of p-adic entire functions on is the union of all of the spaces . It is clear that is dense in , and each function in is of compact support.
For
, let
denote a full collection of coset representatives of
,
i.e.,
Then, , and for any distinct , we have Thus, every non-negative integer k can uniquely be written as , where . Further, a bounded function is said to be a radially-decreasing -majorant of if , and
For
and
, we define the dilation
and translation operators
as follows:
For given
, define the wavelet system:
The wavelet system
is called a wavelet frame, if there exist positive constants
A and
B, such that:
holds for every
, and we call the optimal constants
A and
B the lower frame bound and the upper frame bound, respectively. A tight wavelet frame refers to the case when
, and a Parseval wavelet frame refers to the case when
. On the other hand, if only the right-hand side of the above double inequality holds, then we say
is a Bessel sequence. If both
and
are wavelet frames and for any
, we have the reconstruction formula:
in the
-sense; then, we say that
is a dual wavelet frame of
(and
vice versa), or we simply say that (
) is a pair of dual framelets.
Wavelets and tight wavelet frames are often derived from refinable functions and wavelet masks. A compactly supported function
is called a
p-refinable function, if it satisfies an equation of the type:
where
are complex coefficients. In the Fourier domain, the above refinement equation can be written as:
where:
is a generalized Walsh polynomial, which is called the mask or symbol of the
p-refinable function
ϕ and is of course a
p-adic step function. Observe that
. By letting
in Equations
and
, we obtain
. Since
ϕ is compactly supported and in fact
, therefore
, and hence, as a result,
for all
as
.
Suppose
is a set of
p-MRA functions derived from:
where:
are the generalized Walsh polynomials, called the framelet symbols or wavelet masks. With
as the Walsh polynomials (wavelet masks), we formulate the matrix
as:
The so-called unitary extension principle (UEP) provides a sufficient condition on
, such that the wavelet system
given by Equation (7) constitutes a tight frame for
. It is well known that in order to apply the UEP to derive a wavelet tight frame from a given refinable function, the corresponding refinement mask must satisfy:
Recently, Shah [
18] has given a general procedure for the construction of tight wavelet frames generated by the Walsh polynomials using unitary extension principles as:
Theorem 2.1: Let
be a compactly-supported refinable function, and
. Then, the wavelet system
given by (7) constitutes a Parseval frame in
provided the matrix
as defined in Equation (15) satisfies:
where
3. Periodic Wavelet Frames Related to the Walsh Polynomials
For any
, we define the periodic version of
f as:
Then, it is easy to verify that
is a well-defined locally-integrable function. With the same dilation and translation operators as in Equation (6), we define the periodic wavelet system as:
First, we present an approach for constructing periodic wavelet frames generated by the Walsh polynomials on via the unitary extension principle (UEP). The following theorem is the main result of this section.
Theorem 3.1: Let be the Walsh polynomials given by Equations (12) and (14), and let the wavelet system given by Equation (7) form a Parseval wavelet frame generated by the compactly-supported p-refinable function ϕ. If and have a common radial decreasing -majorant, then the periodic wavelet system given by Equation (18) generates a Parseval wavelet frame for .
We split the proof of Theorem 3.1 into several lemmas.
Lemma 3.2: Suppose that the periodic wavelet system
is as in Theorem 3.1. Then, for any function
and given
, there exists a positive integer
, such that:
Proof: Let
S denote the support of the Walsh–Fourier coefficients
. Then, we have:
Let:
where the Walsh–Fourier coefficients of the above series are given by:
Applying Parseval’s formula to the above Walsh–Fourier series, we obtain:
where
As
S is a finite set, there exists a positive number
N, such that
Hence, there exists
, such that for all
, the elements of
lie in different cosets of
(see [
13]). Thus, the cardinality of
is at most one for each
. Consequently, we have:
Since
, therefore there exists a non-negative integer
, such that:
Let
, then with this choice of
, we obtain:
By using Equation (4), we have:
This completes the proof of Lemma 3.2. ☐
Lemma 3.3: Let
be the refinement mask of a compactly-supported refinable function
ϕ of an MRA, and let
be the wavelet masks. Moreover, if the wavelet system
given by Equation (7) forms a normalized tight wavelet frame for
, then for any
, we have:
Proof: For any
and
, define the linear operators
and
as:
Since
is a dense subset of
, it is sufficient to prove that:
holds for all of the functions
f in
. Therefore, for all
and
, we obtain the following equality by using Parseval’s formula:
By taking advantage of the periodicity of the Walsh polynomial
, we obtain:
where:
Proceeding on similar lines as above, we can have:
Since the UEP condition Equation (17) is equivalent to:
therefore, we have:
and hence, we get the desired result. ☐
Lemma 3.4: Let
be a compactly-supported refinable function with refinement mask
, and let the wavelet system
given by Equation (7) constitute a Parseval wavelet frame for
. Moreover, if
and
have a common radial decreasing
-majorant, then we have:
Proof: For any
and
, we have:
The change of the summation and the integration above is reasonable. In fact, we have:
We can also deduce that the series:
is absolutely convergent. Therefore, the series can be rearranged as follows:
For
, we define:
where
is the characteristic function. Using the fact that
, we have:
Similarly, for each
, we have:
The application of Lemma 3.3 yields:
This completes the proof. ☐
Proof of Theorem 3.1: For any function
and
, we can choose
by Lemma 3.2, such that for all
, we have:
For any
, Lemma 3.4 implies that:
By repeating this argument on
, we obtain:
Letting
, we obtain:
Since
was arbitrary. Therefore, it follows that:
This completes the proof of the Theorem 3.1. ☐
4. Dual Periodic Wavelet Frames Related to the Walsh Polynomials
In this section, we construct dual periodic wavelet frames generated by the Walsh polynomials on using the mixed extension principle (MEP). The following theorem is the main result of this section.
Theorem 4.1: Suppose that ϕ and are two compactly-supported refinable functions, and are the Walsh polynomials. Let and be a pair of dual wavelet frames for generated by the mixed extension principle. Then, and form a pair of dual wavelet frames for .
We need the following lemmas, which are important for the proof of the main result.
Lemma 4.2: The sequences and are both Bessel sequences for .
Proof. To simplify expressions in the proof, we let:
In order to prove that the sequences
and
are both Bessel sequences, we need to find out two positive numbers
, such that for any function
, we have:
For any
, by the Parseval identity of the Walsh–Fourier series, we deduce that:
Since
, so
is bounded on
; therefore, there exists
, such that
. Using Equation (4) and this estimate, Equation (27) reduces to:
Next, we compute
. Using the periodic property of the functions
, that is
, we have:
Since the wavelet system
is a Bessel sequence for
, thus we can deduce that there exists a positive number
, such that:
Combining Equations (28) and (29), we get:
This completes the proof of Lemma 4.2.
Lemma 4.3: If
,
i.e.,
where the sequences
and
have only finitely many non-zero terms. Then, the following formula holds:
Proof: We split the proof of this results into three steps.
Step 1: We rearrange and rewrite the following series:
Since both the functions
ϕ and
are compactly supported, thus it is possible to make a rearrangement in the above series:
For any
, we define:
Since
f and
g are periodic functions, by Equations (31) and (32), we have:
By summing Equation (33) over the set
and noting that
, we have:
Similarly, for each
, we have:
Step 2: For any
, we claim that:
Taking the sum on the R.H.S of Equation (35) over
, we have:
By the Parseval identity of the Walsh–Fourier transform and Equation (13), we deduce that:
Since
are the Walsh polynomials (wavelet masks) associated with given wavelets
and we know each
is bounded periodic on
, therefore we have:
Therefore, the exchange of the integral and the summation is reasonable in the above formula. Again, by the periodicity of wavelet masks
, we infer that:
Since
is an orthonormal basis for
, therefore, by Equations (38) and (39), together with the Parseval identity of the Walsh–Fourier series, we have:
Again, by Equation (36), we deduce that:
By the mixed extension principle (MEP) condition, we have:
Using the Parseval identity of the Walsh–Fourier series, we obtain:
Again, by the Parseval identity of the Walsh–Fourier transform, we have:
By Equations (36) and (40), it follows that:
Using Equations (34) and (35) in the above identity, we obtain:
Since
, when
, we have:
In general, for any
, we have:
Step 3: For
, we have:
Since
f and
g are both periodic, hence there exists a non-negative integer
J, such that:
where
. Again, let:
where:
Therefore, for
, we have:
Hence, we conclude that for
, we have:
Since
, we have:
From Equations (41) and (42), we deduce that:
This completes the proof of the Lemma 4.3. ☐
Proof of Theorem 4.1: By Lemma 4.2, it follows that the sequences and are both Bessel sequences for . By Lemma 4.3, we know that for any , Equation (30) holds. Again, since the set is dense in , it follows that the periodic wavelet systems and constitute a pair of dual frames for . This completes the proof. ☐