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We characterize the orthogonal frames and orthogonal multiwavelet frames in

Wavelets are mathematical functions that take account into the resolutions and the frequencies simultaneously [

The classical MRA scaler wavelets are probably the most important class of orthonormal wavelets. However, the scalar wavelets cannot have the orthogonality, compact support, and symmetry at the same time (except the Haar wavelet). It is a disadvantage for signal processing. Multiwavelets have attracted much attention in the research community, since multiwavelets have more desired properties than any scalar wavelet function, such as orthogonality, short compact support, symmetry, and high approximation order [

Although many compression applications of wavelets use wavelet or multiwavelet bases, the redundant representation offered by wavelet frames has already been put to good use for signal denoising and image compression. In fact, the concept of frame was introduced a long time ago [

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

Let

A collection of elements

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

Let us now fix an arbitrary matrix

Then, we define the multiwavelet frame, the multiwavelet tight frame, the multiwavelet tight frame, and the filter.

We say that

We say that

We say that

We turn to the concept of multiresolution analysis (MRA) in

Let

There exists a function

There is a standard procedure for constructing multiwavelets from a given MRA

In the following, we will borrow some notations from [

Let

Let

Let

A closed subspace

We consider orthogonal frames in a shift-invariant subspace of

For

Let

We define the mixed dual Gramian as

In this section, we present a simple construction of a pair of orthogonal multiwavelet frames from two arbitrarily multiwavelet frames and get some interesting properties about the orthogonal multiwavelet frames. We also show different algorithms for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Firstly, we give a lemma, which has been obtained by Weber in [

Let

From Lemma

Let

Assume that

Let

We now show that the multiwavelet systems generated by

Moreover,

The following results give some properties of the orthogonal frames.

Suppose that

Suppose that

Again by

Suppose that

Similar to the proof in Proposition

Then, we recall a result from [

Suppose

We call

With the above definitions, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Suppose that

For Items (a) and (b), by Lemma

Let us focus on

Likewise, for

The following results show the relationship between a pair of orthogonal MRA multiwavelet frames.

Suppose that

Suppose that

For any

Notice that

Applying Fourier inverse transform on (

From the above result, we get the following equation:

For any

Putting everything together, we have

The following theorem describes a general construction algorithm for orthogonal multiwavelet tight frames.

Suppose

Firstly, we prove that

Similarly,

Now, since the entries of

For orthogonality, according to (

The following proposition is directly related to the construction algorithm in Theorem

If

The proof will follow the notation of Theorem

From the above results, we get the following equation:

Hence,

In this paper, motivated by the notion of orthogonal frames, we present the construction of orthogonal multiwavelet frames in

L. Zhanwei was supported by the Henan Provincial Natural Science Foundation of China (Grant no. 102300410205). H. Guoen was supported by the National Natural Science Foundation of China (Grant no. 10971228).

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