A Short Note on Wavelet Frames Based on FMRA on Local Fields

Abstract

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

1. Introduction

Let be a field and a topological space. Then, is called a local field if both and are locally compact abelian groups, where and denote the additive and multiplicative groups of , respectively. If is any field and is endowed with the discrete topology, then is a local field. Furthermore, if is connected, then is either or . If is not connected, then it is totally disconnected. Hence, by a local field, we mean a field which is locally compact, nondiscrete, and totally disconnected. The -adic fields are examples of local fields. For more details, refer [1]. In the rest of this paper, we use the symbols and to denote the sets of natural and nonnegative integers and integers, respectively.

Let be a local field. Let be the Haar measure on the locally compact abelian group . If and , then is also a Haar measure. Let . We call the absolute value of . Moreover, the map has the following properties: (a) if and only if ; (b) for all ; and (c) for all . Property (c) is called the ultrametric inequality. The set is called the ring of integers in . Define . The set is called the prime ideal in . The prime ideal in is the unique maximal ideal in , and therefore is principal ideal as well as prime ideal. Since the local field is totally disconnected, there exists an element of of maximal absolute value. Let be a fixed element of the maximum absolute value in . Such an element is called a prime element of . Therefore, for such ideal in , we have . As it was proved in [1], the set is compact and open. Hence, is compact and open. Therefore, the residue space is isomorphic to a finite field , where for some prime and .

Let . Then, it can be proved that is a group of units in , and if , then we may write . For a proof of this fact, refer [1]. Moreover, each is a compact subgroup of and usually known as the fractional ideals of . Let be any fixed full set of coset representatives of in ; then, every element can be expressed uniquely as with . Let be a fixed character on that is trivial on but is nontrivial on . Therefore, is a constant on cosets of , so if , then . Suppose that is any character on ; then, clearly, the restriction is also a character on . Therefore, if is a complete list of the distinct coset representative of in , then, as it was proved in [1], the set of distinct characters on is a complete orthonormal system on .

The Fourier transform of a function is defined by

It is noted that

Furthermore, the properties of the Fourier transform on local field are much similar to those on the real line. In particular, Fourier transform is unitary on .

We now impose a natural order on the sequence . We have , where is a -dimensional vector space over the field . We choose a set such that span . For satisfyingwe define

Also, for , we set

This defines for all . In general, it is not true that . However, if , then . Furthermore, it is also easy to verify that if and only if and for a fixed . Hereafter, we use the notation .

Let the local field be of characteristic and be as above. We define a character on as follows:

In 2015, Shah [2] introduced the concept of frame multiresolution analysis (FMRA) on local fields, which can be sought as an extension of multiresolution analysis (MRA) on local fields of positive characteristic. First of all, let us recall the definition of FMRA as given by Shah. Let be a local field of positive characteristic and be a prime element of . A frame multiresolution analysis (FMRA) of is a sequence of closed subspaces of satisfying the following properties:(a)(b) is dense in (c)(d) if and only if (e)There is a function such that forms a frame in

The function is called a frame refinable function. It is noted that the shifts of form a tight frame in the above FMRA. Replacing “a tight frame” in the above by “an orthonormal or a Riesz base” will arrive on the definition of a MRA on local fields of positive characteristic.

A finite family generates a wavelet frame for if there exist positive numbers such that, for all ,where . The largest constant and the smallest constant satisfying the above are called the lower and upper wavelet frame bound, respectively. A wavelet frame is a tight wavelet frame if and are chosen so that , and then the set is called a set of generators for the corresponding tight wavelet frame. Furthermore, the wavelet frame is called a Parseval wavelet frame if , i.e.,and in this case, every function can be written as

A tight wavelet frame is called a FMRA tight frame on local fields with frame bound 1 if . Here, in this article, we are concerned with a minimum-energy wavelet frame which is more restrictive than a FMRA tight frame on local fields of positive characteristic. Here, we recall the definition of minimum-energy wavelet frames on local fields of positive characteristic [3].

Definition 1. Let satisfy , be continuous at 0, and . Suppose that generates the nested closed subspaces . Then, a finite family is called a minimum-energy wavelet frame associated with ifwhere . By the Parseval identity, minimum-energy wavelet frame must be a tight frame for with the frame bound being equal to 1. At the same time, the above equation is equivalent toMotivated and inspired by various constructions of minimum-energy wavelet frames [4–10] and classical wavelet frames on finite fields [11–13], we, in this paper, discuss some constructions of minimum-energy wavelet frames which are based on the frame multiresolution analysis on local fields of positive characteristic. This paper is organized in the following manner. In Section 2, we present some preliminaries for the FMRA and the minimum-energy wavelet frames on local fields of positive characteristic. In Section 3, we present the main results. Here, we provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. We also construct an example to illustrate our algorithm.

2. Notations and Preliminaries

Here, we present some preliminaries for the FMRA and the minimum-energy wavelet frames on local fields of positive characteristic.

From the definition of FMRA on local fields, we know that . Since , there exists a sequence such that

The Fourier transform of (12) yieldswhereis an integral periodic function in , where is the ring of integers in and is often called the refinement symbol of . Observe that . Therefore, by letting in (13) and (14), we obtain .

Consider , with

Equation (15) can be written in the frequency domain aswhereare the integral periodic function in and are called the framelet symbols or wavelet masks.

With , as framelet symbols, we formulate the matrix as

Shah and Debnath [3] proved that if forms a minimum-energy wavelet frame in , then the mask matrix should satisfy certain conditions as follows.

Lemma 1. Suppose that the refinable function and the framelet symbols , satisfy (13)–(17). If is continuous at 0 and generates a sequence of nested closed subspaces , then the following statements are equivalent:(1) is a minimum-energy frame associated with .(2) is an identity matrix of order (19).(3).Lemma 1 gives the necessary and sufficient condition for the existence of the minimum-energy wavelet frames associated with refinable function . However, it is not a good choice to use this theorem to construct the minimum-energy wavelet frames. For convenience, Shah and Debnath [3] presented some conditions in terms of the framelet symbols.

Lemma 2. Let be the refinable function with refinement mask such that is continuous at 0 and . If is the minimum-energy wavelet frame associated with , thenIn this paper, we start with the orthogonal vectors to obtain an explicit construction for the minimum-energy wavelet frame on local fields of positive characteristic.

3. Main Results

From Lemma 1, we know that if we want to obtain a minimum-energy wavelet frame on local fields of positive characteristic, we should find functions whose masks satisfy (19). Note the correlation of the rows of ; we should remove this feature first. For this, we introduce the polyphase decomposition technique. Similar to [3], we writewhere and , are the polyphase decompositions of and , respectively, and all these functions are periodic. Let

Then, we can easily see that , and (19) is equivalent to

The difference of (19) and (25) is that the rows of (25) are irrelevant to one another. Since , we have .

Riesz lemma tells us that there can exist a function such that . When , then , which is the special case of the orthonormal wavelet base on local fields of positive characteristic. With in hand, we can have a vector, in fact, a unit column vector, . Now, we expect the existence of two more unit column vectors and such that

In fact, and form an orthonormal fundamental system of solutions of the linear equation . By straightforward calculation, an orthonormal fundamental system of solutions is

Here, . Here, we notice that if we choose as the first two rows of the following matrixthen is a unitary matrix which implies that the corresponding matrix is a mask matrix. Moreover, the minimum-energy wavelet frame matrix on local fields has the shapewhere is a square matrix which is also unitary.

Given a refinement function , the refinement mask should satisfy , which is the same as to the first column of (26). So, we can select the orthonormal wavelet masks as those of (26). So, all the orthonormal wavelet masks are of the shape

4. Example

Let us consider the case of the Haar wavelet. The refinement function of the Haar wavelet on local fields will be given by , and its refinement mask will be . It can be observed that satisfies , and the corresponding polyphase decompositions are . Here, we notice that the polyphase decompositions of the orthonormal wavelet masks are and . Hence, the wavelet masks are given by .

Data Availability

The data used to support the findings of this study are available upon request to the author.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the UGC-BSR Research Start-Up Grant (no. F.30-498/2019 (BSR)) provided by the UGC, Govt. of India.