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High balanced biorthogonal multiwavelets with symmetry.

1. Introduction

Multiwavelets have been studied extensively in the literature; for example, to mention only a few here, see [1-7] and the references therein. Implementing multiwavelet transform, one will face a serious problem related to the discrepancy of their approximation properties between the function setting and the discrete vector data setting [1, 8-10]. As such, a multiwavelet must be prefiltered or balanced in advance [1, 2, 8-10]. Balanced multiwavelet transform can sparsely process the vector-valued data efficiently. Moreover, they can preserve the polynomial structure of a signal. As such, constructing balanced multiwavelets is of interest for researchers of wavelet analysis. In this paper, we will construct pairs of biorthogonal symmetric multiwavelets with high balanced orders from any orthogonal refinable function vector, which simultaneously has symmetric and antisymmetric components. Before introducing our motivation, let us give some conceptions and notations.

Let [N.sub.0] denote the set of nonnegative integers while [D.sup.j]f(x) denotes the jth derivative of f(x), j [member of] [N.sub.0]. Suppose that a compactly supported d-refinable function vector [phi]: = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T]: R [??] [C.sup.r x 1] satisfies the refinement equation

[phi] = d[summation over (k [member of] Z)][a.sub.k][phi](d x -k), (1)

with the integer d ([greater than or equal to] 2) being regarded to as a dilation factor. We get from (1) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [??]([xi]) = [[summation].sub.k [member of] Z][a.sub.k][e.sup.-ik[xi]] being called the mask symbol of [phi] and Fourier transform [??] of g [member of] [L.sub.1](R) being defined to be [??]([xi]) = [[integral].sub.R]g(x)[e.sup.-ix[xi]]dx, which can be naturally extended to square integrable functions and temper distributions. We say that [??]([xi]) has m sum rules if there exists a 1 x r matrix [??]([xi]): = ([[??].sub.1]([xi]), [[??].sub.2]([xi]), ..., [[??].sub.r]([xi])) of 2[pi]-periodic trigonometric polynomials with [??](0) [not equal to] 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We say that [phi] has m approximation orders if there exists a sequence [{([C.sup.1.sub.k, j], [C.sup.2.sub.k, j], ..., [C.sup.r.sub.k, j])}.sub.k [member of] Z] of 1 x r vectors such that, for any polynomial [x.sub.j], j = 0, ..., m - 1, it can be represented as

[x.sup.j] = [summation over (k [member of] Z)]([C.sup.1.sub.k, j], [C.sup.2.sub.k, j], ..., [C.sup.r.sub.k, j])[phi] (x - k). (3)

It is well known that [phi] has m approximation orders if and only if its mask symbol [??]([xi]) has m sum rules.

Next, let us introduce the relationship between the mask symbol of a refinable function vector and its symmetry. Suppose that [phi] = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] [member of] [([L.sub.2](R)).sup.r] in (1) satisfies

[[phi].sub.j](x) = [[epsilon].sub.j][[phi].sub.j]([t.sub.j] - x), [[epsilon].sub.j] [epsilon] {1, -1}, [t.sub.j] [member of] R. (4)

Then, it is not difficult to check that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the words of element, (5) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [??]([xi]) is the (j, k)-element of [??]([xi]). Suppose [??]([xi]) is an r x r matrix of 2[pi]-periodic trigonometric polynomials, which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and then it is not difficult to check that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will use (7) in the proof of Theorem 3.

Let [phi] = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] and [??] = [([[??].sub.1], ..., [[??].sub.r]).sup.T] be two drefinable function vectors in [([L.sub.2](R)).sup.r]. Their mask symbols are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. They are a pair of biorthogonal refinable function vectors, that is,

<[[phi].sub.l](x - k), [[??].sub.l'](x)> = [[delta].sub.k][[delta].sub.l - l'], (9)

if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where * denotes the transpose conjugate. When [phi] = [??], we call [phi] an orthogonal d-refinable function vector.

Let [phi] = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] be an orthogonal d- refinable function vector. We say that it has k + 1 balanced orders if

[[integral].sub.R][[phi].sub.1](x)[x.sup.j]dx = [[integral].sub.R][[phi].sub.2][(x - [1/r]).sup.j]dx = ... = [[integral].sub.R][[phi].sub.r](x)[(x - [[r - 1]/r]).sup.j]dx, j = 0, ..., [kappa]; [kappa] [member of] [N.sub.0]. (11)

There are a few equivalent definitions of balanced multiwavelets in the literature; see [11]. In [10, 11], the conception of balanced multiwavelets is extended to the case of biorthogonal multiwavelets. Next, we will introduce our motivation. The following remark is necessary.

Remark 1 (a balanced refinable function vector does not has antisymmetric component). Suppose that [phi] satisfies (2) and has k + 1 balanced orders. By (11), we get that

[[integral].sub.R][[phi].sub.1](x)dx = [[integral].sub.R][[phi].sub.2](x)dx = ... = [[integral].sub.R][[phi].sub.r](x) dx. (12)

[??](0) satisfies Condition E; that is, 1 is the simple eigenvalue of [??](0) while the other eigenvalues are strictly smaller than 1 in modulus. It follows from (2) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which leads to the fact that [[integral].sub.R][[phi].sub.l](x)dx [not equal to] 0, l = 1, ..., r, and consequently [phi] has no antisymmetric component.

In [8], Yang and Peng constructed balanced multiwavelets via PTST method. Before introducing our motivation, let us briefly introduce the main idea of PTST. Let [phi] be an orthogonal d-refinable function vector. Construct [[phi].sub.[??]] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being a paraunitary matrix; that is, [??]([xi])[??][([xi]).sup.*] = Ir. It is not difficult to check that [[phi].sub.[??]] is still orthogonal. Some conditions are imposed on [??]([xi]) such that [[phi].sub.[??]] has desired balanced orders [8, Theorem 2, Theorem 4]. However, the balanced [[phi].sub.[??]] can not preserve the symmetry of [phi] if it has antisymmetric component. Next, we will explain the reason, which is the motivation of this paper. Without losing generality, assume that

[[phi].sub.1](x) = [[phi].sub.1]([t.sub.1] - x), ..., [[phi].sub.s](x) = [[phi].sub.s]([t.sub.s] - x), [[phi].sub.j](x) = -[[phi].sub.j]([t.sub.j] - x), (13)

j = s + 1, ..., r, s > 0. Suppose that there exists a paraunitary matrix [??]([xi]): = [([[??].sub.i, j]([xi])).sub.i, j] such that [[phi].sub.[??]] is balanced and all the components of [[phi].sub.[??]] have symmetry. By Remark 1, all the components of [[phi].sub.[??]] are symmetric. Then, it is easy to see that [[??].sub.s + 1, l]([xi]), l = 1, ..., r, are all antisymmetric polynomials, which contradicts the fact that [??](0) is invertible. In conclusion, there does not exist invertible matrix [??]([xi]), such that [[phi].sub.[??]] is simultaneously balanced and symmetric if [phi] has antisymmetric component.

Since many famous refinable function vectors satisfy (13), for example, CL multiwavelets [12], we naturally face the following problem.

How can we construct symmetric multiwavelets with high balanced orders from such ones that have antisymmetric component?

In this paper, we are interested in the case of (13) with s = 1. That is,

[[phi].sub.1](x) = [[phi].sub.1]([t.sub.1] - x), [[phi].sub.2](x)= -[[phi].sub.2]([t.sub.2] - x), ..., [[phi].sub.r](x) = - [[phi].sub.r]([t.sub.r] - x). (14)

In fact, many classical multiwavelets satisfy (14). Readers are referred to [13] for many examples.

We will need the following results related to the canonical form of mask symbols of refinable function vectors.

Lemma 2 (see [2]). Let [phi]: = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] be a compactly supported d-refinable function vectors satisfying (1), (2), and (13). Then, for any n [member of] N, there exists a strongly invertible matrix [??]([xi]); that is, [??]([xi]) and [??][([xi]).sup.-1] are both matrices of 2[pi]-periodic trigonometric polynomials, such that the mask symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

where [P.sub.11]([xi]), [P.sub.12]([xi]), [P.sub.21]([xi]), and [P.sub.22]([xi]) are some 1 x 1, 1 x (r - 1), (r - 1) x 1, and (r - 1) x (r - 1) matrices of 2[pi]-periodic trigonometric polynomials, respectively. Moreover, [[phi].sup.u]: = [([[phi].sup.u.sub.1], [[phi].sup.u.sub.2], ..., [[phi].sup.u.sub.r]).sup.T] constructed via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

also satisfies (13) with [[phi].sub.j](x) being replaced by [[phi].sup.u.sub.j](x), j = 1, ..., r, and [D.sup.k][??](0) = 0, l = 2, ..., r, k = 0, ..., n - 1.

Note. (I) The algorithm for constructing [??]([xi]) in Lemma 2 can be seen in the proof of [2, Theorem 2.5]. (II) There exist a number of orthogonal multiwavelets in the literature, whose mask symbols take the form of (15); just see 12, 13] for some examples.

2. Main Results

Theorem 3. Let [phi]: = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] be an orthogonal compactly supported d-refinable function vector satisfying (1), (2), and (14). Then, for any positive integer n, we can construct an r x r matrix [??]([xi]) of 2[pi]-periodic trigonometric polynomials such that the compactly supported d-refinable [[phi].sub.[??]], which is defined via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

is symmetric and has n balanced orders. Moreover, the drefinable function vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is defined via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

is also symmetric and a dual refinable function vector of [[phi].sub.[??]]. On the other hand, [[phi].sub.[??]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have m + 1 and m - 1 approximation orders, respectively.

Theorem 4. Let [phi] = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] be as in Theorem 3. Moreover, its mask symbol [??]([xi]) takes the form of (15) with m [not equal to] 0. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by (18) is compactly supported.

3. Proof of Main Result and Algorithm for High Balanced Biorthogonal Multiwavelets

Proof of Theorem 3. [??]([xi]) has m sum rules. Of course, it has 1 sum rule. According to Lemma 2, there exists a strongly invertible matrix [??]([xi]), such that the mask symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the property of (15); that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where [P.sup.1.sub.11]([xi]), [P.sup.l.sub.12]([xi]), [P.sup.1.sub.21]([xi]), and [P.sup.1.sub.22]([xi]) are some 1 x 1, 1 x (r - 1), (r - 1) x 1, and (r - 1) x (r - 1) matrices of 2[pi]-periodic trigonometric polynomials, respectively. Define [[phi].sup.u]: = [([[phi].sup.u.sub.1], ..., [[phi].sup.u.sub.r]).sup.T] via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Then, [[phi].sup.u] = [([[phi].sup.u.sub.1], ..., [[phi].sup.u.sub.r]).sup.T] satisfies (13). Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, it is not difficult to check that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

By (21) and (6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Again by (6), we know (22) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

where [[??].sub.j] = [t.sub.j] + [[delta].sub.j - 1] and [phi](x) = [([[phi].sub.1](x), ..., [[phi].sub.r](x)).sup.T] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In other words, compared with f, the components of [??](x) are all symmetric.

From the aspect of two-scale similarity transforms (TST), [??]([xi]) is the singular TST of [??]([xi]). Therefore, by [14, Theorem 3.3.3], [??]([xi]) has m + 1 sum rules. Define a strongly invertible r x r matrix [??]([xi]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

with [[??].sub.jr]([xi]) satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

where [t.sub.j] is as in (23).

Construct [[phi].sub.[??]](x) = [([[phi].sub.[??]1](x), ..., [[phi].sub.[??]r](x)).sup.T] via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Then, by (7) and (8), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Moreover, (25) is equivalent to

[[integral].sub.R][[phi].sub.[??]](x)[x.sup.[mu]]dx = [[integral].sub.R][[phi].sub.[??]2](x)[(x - [1/r]).sup.[mu]]dx = ... = [[integral].sub.R][[pi].sub.[??]r](x)[(x - [[r - 1]/r]).sup.[mu]]dx, [mu] = 0, ..., n - 1. (28)

That is, [[phi].sub.[??]] has n balanced orders. Select [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, we will prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is defined via (18), is a dual refinable function vector of [[phi].sub.[??]]. In fact,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

which together with

[(d - 1).summation over (l = 0)][??]([xi] + [2[pi]l/d])[??][([xi] + [2[pi]l/d]).sup.*] = [I.sub.r] (30)

leads to the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a dual refinable function vector of [[phi].sub.[??]]. By (7) and (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

from which we deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

That is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From the aspect of two-scale similarity transforms (TST), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the singular inverse TST of [??]([xi]). Therefore, by [14, Theorem 3.3.3], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has m - 1 sum rules.

Next, we summarize the process of constructing [[phi].sub.[??]] in Theorem 3.

Algorithm 5. Let [phi] be as in Theorem 3. Then, based on [2], [??]([xi]) can be constructed through the following steps.

Step 1. By Theorem 3, construct a strongly invertible r x r matrix [??]/([xi]) of 2[pi]-periodic trigonometric polynomials such that the mask symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] takes the property of (19). Moreover, [??]([xi]) can generate a d-refinable function vector, whose symmetry takes the form of (14).

Step 2. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, [??]([xi]) takes the form of (21). Compared with [??], all the components of the d-refinable function vector [??](x) = [([[??].sub.1](x), ..., [[??].sub.r](x)).sup.T], which is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], satisfy (23).

Step 3. Define a strongly invertible r x r matrix [??]([xi]) as in (24), with [[??].sub.jr]([xi]) satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (25) holds.

Step 4. Construct [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, [[phi].sub.[??]] has n balanced orders.

Step 5. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] via (18). Then, it is the dual symmetric refinable function vector of [[phi].sub.[??]].

Proof of Theorem 4. If the mask symbol of [phi] = [([[phi].sub.1], ..., [[phi].sub.r]).sup.T] in Theorem 3 takes the form of (15), then the dual mask symbol of [??]([xi]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

The transform matrix [??]([xi]) in Step 3 of Algorithm 5 is strongly invertible. Therefore, the mask symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a matrix of 2[pi]-periodic trigonometric polynomials. That is, the dual refinable function vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is compactly supported.

4. High Balanced Biorthogonal Multiwavelets from CL Multiwavelets

In [12], Chui and Lian constructed an orthogonal refinable function vector [phi](x) = [([[phi].sub.1](x), [[phi].sub.2](x)).sup.T], which satisfies the following refinement equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

It is easy to see that [??]([xi]) takes the form of (15) with m = 1 and n = 1. Specifically, straightforward observation gives us that [P.sub.11]([xi]) = ((10 - 3[square root of 10])/80)(1 + (38 + 12[square root of 10])[e.sup.-i[xi]] + [e.sup.- i2[xi]]), [P.sub.12]([xi]) = ((5[square root of 6] - 2[square root of 15])/80)(1 - [e.sup.-2i[xi]]), and [P.sub.21]([xi]) = ((5[square root of 6] - 3[square root of 15])/80)(1 - 10(3 + [square root of 10])[e.sup.-i[xi]] + [e.sup.-2[xi]]). Moreover, by [12], [??]([xi]) has 3 sum rules. See Figure 1 for the graphs of [[phi].sub.1](x) and [[phi].sub.2](x). Select [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Then, by Theorem 3, the components of [phi](x) = [([[phi].sub.1](x), [[phi].sub.2](x)).sup.T], satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are all symmetric. [??]([xi]) has 4 sum rules. By [3], we found out that the Sobolev exponent of [??] is 2.4408. According to Theorem 3, the dual refinable function vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

[??]([xi]) has 2 sum rules.

Next, we will construct balanced symmetric refinable function vectors and their dual refinable function vectors. We compute from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that

[??](0) = [(-4.21498, 1).sup.T], D[??](0) = [(8.42996i, -1.5i).sup.T], [D.sup.2][??](0) = [(17.44843, -2.24893).sup.T]. (40)

We will find a 2[pi]-periodic trigonometric polynomial and construct [[phi].sub.[??]] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

such that [[phi].sub.[??]] is balanced and symmetric.

By (25), if [??]([xi]) satisfies [??](0) = 5.21498, then [[phi].sub.[??]] defined in (41) has 1 balanced order. Take [??]([xi]) = 5.21498(1 + [e.sup.-i[xi]]/2; for example, its graph is illustrated in Figures 2(a) and 2(b).

By (25), if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

then [[phi].sub.[??]] defined in (41) has 2 balanced orders. For this, construct

[??]([xi]) = (1 + [e.sup.-i[xi]]) x [0.30374 ([e.sup.-i[xi]] + [e.sup.i[xi]]) + ([e.sup.-i2[xi]] + [e.sup.i2[xi]])] (44)

such that (43) is satisfied. Now, the graph of [[phi].sub.[??]], given by (41) with [??]([xi]) being as in (44), is shown in Figures 2(c) and 2(d).

By (25), select [??]([xi]), such that [??]([xi]) = [e.sup.- i[xi]][bar.[??]([xi])] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Construct [??]([xi]) = (1 + [e.sup.-i[xi]])(([e.sup.-i[xi]] + [e.sup.i[xi]]) + 0.782280([e.sup.-i2[xi]] + [e.sup.i2[xi]]) - 00.478534([e.sup.-i3[xi]] + [e.sup.i3[xi]]) such that (45) is satisfied. Then, [[phi].sub.[??]] has 3 balanced orders and is symmetric. See Figures 2(e) and 2(f) for its graph.

According to Algorithm 5, the mask symbol of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the dual refinable function vector of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].By [3], we found that the Sobolev exponent of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is 0.4418.

Moreover, for the pair of dual refinable function vectors, one can still get the dual multiwavelets. More specifically, suppose that [??]([xi]) is the mask symbol of [??]([xi]). Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

are the mask symbols of their corresponding multiwavelets, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

5. Future work

In this paper, from an orthogonal refinable function vector [phi], which has antisymmetric component and is not balanced, we give an algorithm for constructing a pair of symmetric biorthogonal refinable function vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [[phi].sub.[??]] has high balanced orders. When compared with [phi], the sum rule of [[phi].sub.[??]] increases by one while that of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] decreases by one. Therefore, the Sobolev smoothness of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] may be smaller than that of [[phi].sub.[??]]. In future, based on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we will study how to construct a new symmetric and smooth dual refinable function vector of [[phi].sub.[??]].

http://dx.doi.org/10.1155/2014/154269

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the two reviewers for their valuable suggestions which improve the paper a lot. The paper is supported by the Natural Science Foundation of Guangxi Province (no. 2013GXNSFBA019010), Natural Science Foundation of China (nos. 11126343, 11461002), and The Midwest Universities Comprehensive Strength Promotion Project of Guangxi University.

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Youfa Li, (1) Shouzhi Yang, (2) Yanfeng Shen, (3) and Gengrong Zhang (1)

(1) College of Mathematics and Information Sciences, Guangxi University, Nanning 530004, China

(2) Department of Mathematics, Shantou University, Shantou 515063, China

(3) Department of Mathematics, Dezhou University, Dezhou 253023, China

Correspondence should be addressed to Yanfeng Shen; shyf8584@163.com

Received 10 June 2014; Accepted 28 August 2014; Published 10 November 2014

Academic Editor: Thanh Tran
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