Journal of Operators
Volume 2015 (2015), Article ID 216060, 8 pages
http://dx.doi.org/10.1155/2015/216060

Research Article

Frame Multiresolution Analysis on Local Fields of Positive Characteristic

Firdous A. Shah

Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir 192101, India

Received 21 August 2014; Accepted 12 January 2015

Academic Editor: Gerd Teschke

Copyright © 2015 Firdous A. Shah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace of has a frame, a collection of translates of the scaling function of the form , where is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field of positive characteristic using the shift-invariant space theory.

1. Introduction

Multiresolution analysis is considered as the heart of wavelet theory. The concept of multiresolution analysis provides a natural framework for understanding and constructing discrete wavelet systems. A multiresolution analysis is an increasing family of closed subspaces of such that and is dense in which satisfies if and only if . Furthermore, there exists an element such that the collection of integer translates of function , , represents a complete orthonormal system for . The function is called the scaling function or the father wavelet. The concept of multiresolution analysis has been extended in various ways in recent years. These concepts are generalized to , to lattices different from , allowing the subspaces of multiresolution analysis to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer or by an expansive matrix as long as (see [1, 2]).

On the other hand, this elegant tool for the construction of wavelet bases has been extensively studied by several authors on the various spaces, namely, Cantor dyadic groups [3], locally compact Abelian groups [4], -adic fields [5], zero-dimensional groups [6], and Vilenkin groups [7]. Recently, R. L. Benedetto and J. J. Benedetto [8] developed a wavelet theory for local fields and related groups. They did not develop the multiresolution analysis (MRA) approach; their method is based on the theory of wavelet sets. The local fields are essentially of two types: zero and positive characteristic (excluding the connected local fields and ). Examples of local fields of characteristic zero include the -adic field whereas local fields of positive characteristic are the Cantor dyadic group and the Vilenkin -groups. The structures and metrics of the local fields of zero and positive characteristic are similar, but their wavelet and MRA theory are quite different. The concept of multiresolution analysis on a local field of positive characteristic was introduced by Jiang et al. [9]. They pointed out a method for constructing orthogonal wavelets on local field with a constant generating sequence. Subsequently, tight wavelet frames on local fields of positive characteristic were constructed by Shah and Debnath [10] using extension principles. As far as the characterization of wavelets on local fields is concerned, Behera and Jahan [11] have given the characterization of all wavelets associated with multiresolution analysis on local field based on results on affine and quasiaffine frames. Recently, Shah and Abdullah [12] have introduced the notion of nonuniform multiresolution analysis on local field of positive characteristic and obtained the necessary and sufficient condition for a function to generate a nonuniform multiresolution analysis on local fields. More results in this direction can also be found in [13, 14] and the references therein.

Since the use of multiresolution analysis has proven to be a very efficient tool in wavelet theory mainly because of its simplicity, it is of interest to try to generalize this notion as much as possible while preserving its connection with wavelet analysis. In this connection, Benedetto and Li [15] considered the dyadic semiorthogonal frame multiresolution analysis of with a single scaling function and successfully applied the theory in the analysis of narrow band signals. The characterization of the dyadic semiorthogonal frame multiresolution analysis with a single scaling function admitting a single frame wavelet whose dyadic dilations of the integer translates form a frame for was obtained independently by Benedetto and Treiber by a direct method [16] and by Kim and Lim by using the theory of shift-invariant spaces [17]. Later on, Yu [18] extended the results of Benedetto and Li’s theory of FMRA to higher dimensions with arbitrary integral expansive matrix dilations and has established the necessary and sufficient conditions to characterize semiorthogonal multiresolution analysis frames for .

In this paper, we introduce the notion of frame multiresolution analysis (FMRA) on local field of positive characteristic by extending the above described methods. We first investigate the properties of multiresolution subspaces, which will provide the quantitative criteria for the construction of FMRA on local fields of positive characteristic. We also show that the scaling property of an FMRA also holds for the wavelet subspaces and that the space can be decomposed into the orthogonal sum of these wavelet subspaces. Finally, we study the characterization of wavelet frames associated with FMRA on local field of positive characteristic using the shift-invariant space theory.

The paper is organized as follows. In Section 2, we discuss some preliminary facts about local fields of positive characteristic including the definition of a frame. The notion of frame multiresolution analysis of is introduced in Section 3 and its quantitative criteria are given by means of Theorem 12. In Section 4, we establish a complete characterization of wavelet frames generated by a finite number of mother wavelets on local field of positive characteristic.

2. Preliminaries on Local Fields

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. Further, 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. More details are referred to in [19, 20]. In the rest of this paper, we use , and to denote the sets of natural and nonnegative integers and integers, respectively.

Let be a fixed local field. Then, there is an integer , where is a fixed prime element of and is a positive integer, and a norm on such that for all we have and for each we get for some integer . This norm is non-Archimedean; that is, for all and whenever . Let be the Haar measure on the locally compact, topological group . This measure is normalized so that , where is 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 hence as a result is both principal and prime. Therefore, for such an ideal in , we have .

Let . Then, it is easy to verify that is a group of units in and if , then we may write , . Moreover, each is a compact subgroup of and is known as the fractional ideals of (see [19]). 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 constant on cosets of , implying that if , then for . Suppose that is any character on ; then, clearly the restriction is also a character on . Therefore, if is a complete list of distinct coset representatives of in , then, as it was proved in [20], 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 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 . Since , where is a -dimensional vector space over the field (see [20]), we choose a set such that span . For such that , we have

Define

For and , , we write

such that

Also, for and , we have

Further, it is 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 let be as above. We define a character on as follows:

Definition 1. Let be a separable Hilbert space. A sequence in is called a for if there exist constants and with such that

The largest constant and the smallest constant satisfying (9) are called the upper and the lower frame bound, respectively. A frame is said to be tight if it is possible to choose and a frame is said to be exact if it ceases to be a frame when any one of its elements is removed. An exact frame is also known as a Riesz basis.

The following theorem gives us an elementary characterization of frames.

Theorem 2 (see [15]). A sequence in a Hilbert space is a frame for if and only if there exists a sequence with , , such that

and , for every .

For and , we define the dilation operator and the translation operator as follows:

Our study uses the theory of shift-invariant spaces developed in [21, 22] and the references therein. A closed subspace of is said to be shift-invariant if whenever and . A closed shift-invariant subspace of is said to be generated by if . The cardinality of the smallest generating set for is called the length of which is denoted by . If finite, then is called a finite shift-invariant space (FSI) and if , then is called a principal shift-invariant space (PSI). Moreover, the spectrum of a shift-invariant space is defined to be

where .

3. Frame Multiresolution Analysis on Local Fields

We first introduce the notion of a frame multiresolution analysis (FMRA) of .

Definition 3. Let be a local field of positive characteristic and let be a prime element of . A frame multiresolution analysis of is a sequence of closed subspaces of satisfying the following properties:

(a) for all ;(b) is dense in and ;(c) if and only if for all ;(d)the function lying in implies that the collection , for all ;(e)the sequence is a frame for the subspace .

The function is known as the scaling function while the subspaces ’s are known as approximation spaces or multiresolution subspaces. A frame multiresolution analysis is said to be nonexact and, respectively, exact if the frame for the subspace is nonexact and, respectively, exact. In multiresolution analysis studied in [9], the frame condition is replaced by that of an orthonormal basis or an exact frame.

Next, we establish several properties of multiresolution subspaces that will help in the construction of frame multiresolution analysis on local field of positive characteristic. The following proposition shows that, for every , the sequence , where

is a frame for .

Proposition 4. Let be a frame for and

Then, the sequence defined in (13) is a frame for with the same bounds as those for .

Proof. For any , we have

Since is a frame for , therefore we have

This completes the proof of the proposition.

We now characterize all functions of FSI space by virtue of its Fourier transforms.

Proposition 5. Let be a frame for its closed linear , where . Then, lies in if and only if there exist integral periodic functions , , such that

Proof. Since the system is a frame for , then, by Theorem 2, there exists a sequence , for , such that

Taking Fourier transform on both sides of (20), we obtain

where are the integral periodic functions in . The converse is established by taking as above and applying the inverse Fourier transform on both sides of (19).

We now study some properties of the multiresolution subspaces of the form (14) by means of the Fourier transform.

Proposition 6. Let be a frame for and, for , define by (14). Then, for any function , there exists periodic function such that

Proof. By the definition of , it follows that . By Proposition 5, there exists a periodic function such that lies in .

The following theorem establishes a sufficient condition to ensure that the nesting property holds for the subspaces ’s.

Theorem 7. Let be a frame for and, for , define by (14). Assume that there exists a periodic function such that

Then, , for every .

Proof. Given any , there exists a sequence such that

Let and let . Then, clearly lies in as lies in . Therefore, by Parseval’s identity, there exists a sequence such that lies in .
Taking Fourier transform of (24) and using assumption (23), we obtain

By implementing inverse Fourier transform to (25), we have

Using Proposition 4, we observe that . Moreover, it is easy to verify that the function in (23) is not unique.

The following theorem is the converse to Theorem 7.

Theorem 8. Let be a frame for and, for , define by (14). Assume that and . Then, there exists periodic function such that (23) holds.

Proof. Since is a frame for , therefore there exist positive constants and such that

Since , we have . By Proposition 6, there exists a periodic function such that

Therefore, we have

Let and be a periodic function such that , a.e. on , and is bounded on by a positive constant . Then, it follows from the above fact that is not unique so that (29) also holds for ; that is,

Taking , where and , we have

Summing up (31) for all and , we have

which is equivalent to

or

Note that a.e. and, hence, (34) becomes

This implies that, for almost every and , we have

Also, if , then and if , then we may assume that . Thus, for almost every and , we have

Hence, is essentially bounded on . This proves the theorem completely.

The following two propositions are proved in [23].

Proposition 9. Suppose and, for each , define by (14) such that . Assume that on a neighborhood of zero. Then, the union is dense in .

Proposition 10. Let and define . For each , define by (14). Then, one has .

Lemma 11. Let be the family of subspaces defined by (14) with , for each . Suppose is a nonzero function with . Then, for every is a proper subspace of .

Proof. Suppose that for some . Let ; then, for any given , we have . Since , therefore lies in and . Hence, . By Proposition 10, it follows that , which is a contradiction.

Combining all our results so far, we have the following theorem.

Theorem 12. Let and define . For each , define by (14) and . Suppose that the following hold:

(i) a.e. on ,(ii)there exists a periodic function such that

(iii), a.e. on a neighborhood of zero.

Then, defines a frame multiresolution analysis of .

Proof. Since is a shift-invariant subspace of , therefore the system forms a frame for with frame bounds and . By Theorem 7 and Lemma 11, it follows that , for every . Hence, by the definition of lies in if and only if lies in , while lies in if and only if lies in . Thus, lies in if and only if lies in . Moreover, by assumption (iii) and Proposition 10, it follows that is dense in and . Thus, the sequence satisfies all the conditions to be a frame multiresolution analysis of .

In order to construct wavelet frames associated with frame multiresolution analysis on local fields of positive characteristic, we introduce the orthogonal complement subspaces of in . It is easy to verify that the sequence of subspaces also satisfies the scaling property; that is,

Theorem 13. Let be an increasing sequence of closed subspaces of such that is dense in and . Let be the orthogonal complement of in , for each . Then, the subspaces are pairwise orthogonal and

Proof. Assume that ; then, , for any as . Let be the orthogonal projection operators from onto ; then, ,  , and . Therefore, for any , we have

Thus, the result of the direct sum follows since is the orthogonal projector from onto .

4. Characterization of Wavelet Frames on Local Fields

In this section, we give the characterization of wavelet frames associated with frame multiresolution analysis on local fields of positive characteristic. First, we will characterize the existence of a function in , where is the orthogonal complement of in , by virtue of the analysis filters and , defined as in Section 3.

Theorem 14. Let be a periodic function associated with the frame multiresolution analysis satisfying the condition (23). Define as the orthogonal complement of in . Let such that

where is a periodic function in . Then, lies in if and only if

Proof. We note that lies in if and only if

Define

Then, it is easy to verify that lies in by using Monotonic Convergence Theorem and the Plancherel Theorem as

For a fixed , we define as

Then, in view of (23) and (42), we have

Using Monotonic Convergence Theorem and the Cauchy-Schwartz inequality, we obtain

Hence,

Therefore, there exists a subsequence such that

Hence,

Using (50) and the Dominated Convergence Theorem, we have, for all ,

Consequently, , a.e., is the necessary and sufficient condition for (44) to hold for all .

Lemma 15. Let be a sequence of pairwise orthogonal closed subspaces of such that . Then, for every , there exist , , such that . Furthermore,

Proof. For any arbitrary function , we have

where , for each . Moreover, for a fixed , we have

Since the norm is continuous, therefore the desired result is obtained by taking on both sides of the above equality.

Theorem 16. Let be the scaling function for a frame multiresolution analysis and suppose that is the orthogonal complement of in . Let . Then, the collection

constitutes a wavelet frame for with frame bounds and if and only if

forms a frame for with frame bounds and .

Proof. Suppose that the system given by (57) is a wavelet frame for with bounds and . Then, it follows from (39) that the family of functions lies in , for , , and .
By applying Theorem 13 to an arbitrary function , we have

Using the frame property of the system , we have

and it follows that the collection is a frame for .
Conversely, suppose that the collection is a frame for with bounds and . For any fixed and , we have from (39) that . Moreover, by making use of the fact that

we have

Thus, for a given , the collection constitutes a frame for with frame bounds and .
Let be an arbitrary function in ; then, by Theorem 13 and Lemma 15, there exist such that

Therefore, we have

Using (62), we obtain

Combining (64), (65), and Lemma 15, we have

This completes the proof of the theorem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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