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We characterize all generalized lowpass filters and multiresolution analysis(
MRA) Parseval frame wavelets in

Wavelet theory has been studied extensively in both theory and applications since 1980’s (see [

The classical MRA wavelets are probably the most important class of orthonormal wavelets. Because they guarantee the existence of fast implementation algorithm, many of the famous examples often used in applications belong to this class. However, there are useful filters, such as

A tight wavelet frame is a generalization of an orthonormal wavelet basis by introducing redundancy into a wavelet system [

In [

In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets (PFWs) in

Let us now describe the organization of the material that follows. Section

Let us now establish some basic notations.

We denote by

We use the Fourier transform in the form

For

The Lebesgue measure of a set

Then we introduce some notations and the existing results about expanding matrices.

Let

The following elementary lemma [

Let

Our standard example that will be frequently used is the quincunx matrix

In this paper, we will work with two families of unitary operators on

Let us now fix an arbitrary matrix

Let us recall the definition of a Parseval frame and a Parseval frame wavelet.

We say that a countable family

We say that

Then we recall a result from [

Let

In the following, we will give some definitions which will be used in this paper. In fact, they are some generalizations of the notations in [

A measurable

We will denote by

A function

Notice that

Suppose that

Consequently, if

For

Then, we will give the definition of MRA PFW.

A PFW

Let us conclude this introductory section by noting that many of the results that follow can be proved for dilations by expanding integer matrices with arbitrary determinant. Some of these extensions are obtained easily with essentially the same proofs; others require subtler and more involved arguments. But, for the sake of simplicity, we restrict ourselves to the class

The main purpose of this section is to study the pseudo-scaling functions, the generalized filters, and the MRA PFWs in

In the following we firstly give several lemmas in order to prove our main results.

Suppose that

By (

If

Assuming that

It follows that for a.e.

Then, we will give a characterization of the generalized lowpass filter.

Suppose

Since

The following theorem provides a way of constructing MRA PFWs from generalized lowpass filters.

In order to obtain main result in this part, we firstly introduce a result in [

Let

Suppose that

Suppose that

Thus,

For any unimodular function

From the fact that

Let us now prove that the function

Let us fix

To compute the first term on the right-hand side of (

To compute the second term on the right-hand side of (

This shows that the expression in (

In this section, we will describe the multiplier classes associated with PFWs in

A PFW multiplier

An MRA PFW multiplier is a function

A pseudo-scaling function multiplier is a function

A generalized lowpass filter multiplier is a function

At first, we will obtain a property of PFW multiplier.

If a measurable function

Let

Let

In particular, for a.e.

The next theorem gives a characterization of PFW multiplier in

If a measurable function

PFW is characterized as an element of

Let

If

By

By the equalities (

Since the above equation of the right-hand side is 0 by (

Hence, from Lemma

It is proved in [

Let

A measurable function

(if) Let

Let

Let

It is obvious that

Let

(only if) The Haar wavelet is, in particular, an MRA TFW; we can then proceed as in the proof of Property

Thus, we complete the proof.

The next theorem characterizes the class of the generalized lowpass filter multipliers. Note that the generalized lowpass filters are the functions

A measurable function

(if) Suppose that

Let

(only if) We proceed similarly in the proof of the two previous theorems.

Let

Let

Applying

This implies that

In a conclusion,

The above results provide description of PFW, MRA PFW, and generalized lowpass filter multipliers. These classes are identical with the respective multiplier classes of wavelets. This fact is basically a consequence of the fact that all of these multiplier operations necessarily preserve the

Let us introduce a notation for the next theorem. For a measurable function

The following result shows that the situation for pseudo-scaling function multipliers is completely different from others.

If

(if) Let

Let

Using condition (1), we see that

Let us now examine the scale equation

We claim that there exists

If

If

The condition (2) implies that

Let us consider

For any

In (a) case,

In (b) case we extend the definition of

In (c) case, we let

We have thus extended

Clearly,

(only if) Let

This establishes (2), since the right-hand side is

Let

Let us define

Now we extend

From the definitions of the functions

Finally, we define

However, from [

In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets in

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions that lead to a significant improvement of their manuscript. This work was supported by the National Natural Science Foundation of China (no. 60802061) and the Natural Science Foundation for the Education Department of Henan Province of China (no. 2008B510001).