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Construction of refinable function vector via GTST
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Applied Mathematics and Computation 194 (2007) 425–430 www.elsevier.com/locate/amc
Construction of refinable function vector via GTST Xie Changzhen *, Yang Shouzhi Department of Mathematics, Shantou University, Shantou 515063, PR China
Abstract The concept of general two-scale similarity transform (GTST) is introduced. By applying GTST, we present an algorithm of constructing a class of compactly supported refinable function vector. In addition, we prove that GTST can preserve the symmetry of the given a refinable function vector, and give a sufficient condition that the corresponding transform matrices M(z) and N(z) satisfy when GTST can preserve symmetry. Some examples are given. 2007 Published by Elsevier Inc. Keywords: GTST; Refinable function vector; Symmetry
1. Introduction Several types of uniwavelets are constructed based on multiresolution analysis, such as Daubechies’ orthogonal wavelets [1,2] and semi-orthogonal spline wavelets by Chui and Wang [3]. However, multiwavelets can have some features that uniwavelets cannot. Thus, multiwavelets provide interesting applications in signal processing and some other fields. In recent years, multiscaling functions and multiwavelets have been studied extensively [4–13]. Goodman et al. [4] established a characterization of multiscaling functions and their corresponding multiwavelets. Chui and Wang [5] introduced semi-orthogonal spline multiwavelets. Examples of cubic and quintic finite elements and their corresponding multiwavelets were studied by Strang and Strela [6]. Geronimo et al. [7] used fractal interpolation to construct orthogonal multiscaling functions, and their corresponding multiwavelets were given in [8]. In [9], Donovan, Geronimo, and Hardin showed that there exist compactly supported orthogonal polynomial spline multiscaling functions with arbitrarily high regularity. Are there any easier methods to construct refinable function vector? Can refinable function vector be constructed based on scalar refinable function? As you know, the two-scale similarity transform (TST) is a new, nonobvious construction for multiwavelets [12]. Regular TST have some applications in verifying and imposing symmetry conditions. In this paper, by means of idea of Strela [12], we introduce the concept of GTST, and apply GTST to construct a class of compactly supported refinable function vector.
*
Corresponding author. E-mail address: [email protected] (C. Xie).
0096-3003/$ - see front matter 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2006.12.088
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2. GTST T
Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be a refinable function vector, satisfying the following two-scale matrix equation X UðxÞ ¼ P k Uð2x kÞ; ð1Þ k2Z
where some r · r matrices {Pk} called the two-scale matrix sequence. If r = 1, {Pk} are real number sequence and U(x) = /1(x). In this case we say that (1) is a scalar dilation refinement equation. Solutions of Eq. (1) have been extensively studied in recent years [13–18]. It turned out that it is more conb venient to work with the Fourier transform UðwÞ that with the refinable function U(x) itself. By taking Fourier transform on both sides of (1), we have b b w ; z ¼ eiw=2 ; UðwÞ ¼ P ðzÞ U ð2Þ 2 P where P ðzÞ ¼ 12 k2Z P k zk called two-scale matrix symbol corresponding to U(x). In the scalar case P(z) is a trigonometric polynomial. In the multi case P(z) becomes a matrix trigonometric polynomial. We called that U(x) generates a space V0 if V0 consists of all finitely linear combination of integer translates of entries of U(x). Assume that U(x) generates a space V0. Then for any compactly supported functions w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ in V0, there exists a finitely many nonzero matrices Mk of size s · r such that X T WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ ¼ M k Uðx kÞ: ð3Þ k2Z
In terms of Fourier transform, we have b b WðwÞ ¼ MðzÞ UðwÞ;
P
ð4Þ k
where M(z) denotes the s · r matrix of Laurent polynomials, i.e., MðzÞ ¼ k2Z M k z . Specially, in the case of r = s, M(z) is a square matrix. A square matrix M(z) is said to be invertible if detðMðzÞÞ is a monomial of z. That is, detðMðzÞÞ ¼ kzm for a scalar k 5 0 and an integer m 2 Z. It is clear that if M(z) is invertible, W(x) generates the same space S. In this case, we have 1 b b b b Wð2wÞ ¼ Mðz2 Þ Uð2wÞ ¼ Mðz2 ÞP ðzÞ UðwÞ ¼ Mðz2 ÞP ðzÞMðzÞ WðwÞ:
ð5Þ
This means that W(x) is an new refinable function vector with two-scale matrix symbol 1
QðzÞ ¼ Mðz2 ÞP ðzÞMðzÞ :
ð6Þ
The case of r = s is well studied by Strela [12]. Definition 1. Assume M(z) of size r · r is a trigonometric matrix polynomial which is invertible. Q(z) is a regular TST of P(z) if QðzÞ ¼ Mðz2 ÞP ðzÞMðzÞ1 :
ð7Þ
e QðzÞ is a regular inverse TST(ITST) of P(z) if e QðzÞ ¼ Mðz2 Þ1 P ðzÞMðzÞ:
ð8Þ
Regular TST have some applications in verifying and imposing symmetry conditions. See [12] for details. By means of idea of Strela [12], we present the concept of GTST. Definition 2. Assume M(z) of size s · r is a trigonometric matrix polynomial, and N(z) of size r · s is a trigonometric matrix polynomial which satisfy N(z)M(z) = Ir. Q(z) of size s · s is a regular GTST of P(z) if QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ:
ð9Þ
C. Xie, S. Yang / Applied Mathematics and Computation 194 (2007) 425–430
427
T
Theorem 1. Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be a refinable function vector with two-scale matrix symbol P(z) of size r · r. Assume M(z) of size s · r is a trigonometric matrix polynomial, and N(z) of size r · s is a trigonometric matrix polynomial which satisfy N(z)M(z) = Ir. Let Q(z) of size s · s be a regular GTST of P(z) with T transform matrices M(z) and N(z). Then WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ generated by Q(z) is an new refinable function vector with multiplicity s, and has the same approximation order as U(x). Proof. Construct function vector WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞT as follows: X M k Uðx kÞ: WðxÞ ¼
ð10Þ
k2Z
By taking Fourier transform the both sides of (10), we have b b WðwÞ ¼ MðzÞ UðwÞ:
ð11Þ
Hence b b b b ¼ Mðz2 ÞP ðzÞ UðwÞ ¼ Mðz2 ÞP ðzÞN ðzÞ WðwÞ: Wð2wÞ ¼ Mðz2 Þ Uð2wÞ
ð12Þ
This means that W(x) is also refinable function with multiplicity s. Its two-scale matrix symbol QðzÞ ¼ b b Mðz2 ÞP ðzÞN ðzÞ. By (11), it is clear that UðwÞ ¼ N ðzÞ WðwÞ. That is, X UðxÞ ¼ N ‘ Wðx ‘Þ: ð13Þ ‘2Z
Suppose that U(x) has approximation order p P 1. Hence, there exists a set of row vectors fujk g R1r that satisfy X j uk Uðx kÞ; j ¼ 0; 1; . . . ; p 1: xj ¼ k2Z
By (13), we have XX j xj ¼ uk N ‘k Wðx ‘Þ; j ¼ 0; 1; . . . ; p 1: k2Z
‘2Z
The above implies that W(x) has also approximation order p. This completes the proof of Theorem 1. Let /(x) be a scalar refinable function. Hence, there exists sequence {pk} such that X pk /ð2x kÞ: ð14Þ /ðxÞ ¼ k2Z
where pðzÞ ¼ 12 Define A0 ðzÞ ¼
P
k2Z p k z
k
1X p zk ; 2 k2Z 2k
called two-scale sequence symbol corresponding to /(x).
A1 ðzÞ ¼
1X p zk : 2 k2Z 2kþ1
ð15Þ
It follows that pðzÞ ¼ A0 ðz2 Þ þ zA1 ðz2 Þ:
ð16Þ
Theorem 2. There exist two Laurent polynomials B0(z) and B1(z) of degree 6n such that A0 ðzÞB0 ðzÞ þ A1 ðzÞB1 ðzÞ ¼ 1:
ð17Þ
Proof. Recalling (16), we see that 1 1 A0 ðz2 Þ ¼ ½pðzÞ þ pðzÞ; zA1 ðz2 Þ ¼ ½pðzÞ pðzÞ: 2 2
ð18Þ
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Since p(z) is a two-scale sequence symbol, p(z) and p(z) have no common zeros (see [1]). Hence A0(z) and A1(z) have no common zeros. By applying Bezout Theorem, there exist two Laurent polynomials B0(z) and B1(z) of degree 6n such that B0(z) and B1(z) satisfies (17). This completes the proof of our claim. Applying the above A0(z), A1(z), B0(z) and, B1(z), we can construct M(z), N(z) as follows: ( T Mðz2 Þ ¼ ½A0 ðz2 Þ þ zA1 ðz2 Þ; B1 ðz2 Þ þ zB0 ðz2 Þ ; ð19Þ N ðz2 Þ ¼ ½B0 ðz2 Þ; A1 ðz2 Þ: It is easy to verify that N ðzÞMðzÞ ¼ 1:
ð20Þ
Next, we discuss the case of r = s. If r = s, then GTST becomes TST. Specially, if N ðzÞMðzÞ ¼ I r and N ðzÞ ¼ MðzÞ , then N(z)(or M(z)) is a paraunitary matrix. h T
Theorem 3. Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be an orthogonal refinable function vector with two-scale matrix symbol P(z) of size r · r. Assume r · r matrix M(z) is a paraunitary matrix. Let Q(z) of size r · r be a regular GTST of P(z) with transform matrices M(z) and MðzÞ . Then WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; wr ðxÞT generated by Q(z) is also an orthogonal refinable function vector, and has the same approximation order as U(x). Proof. In Theorem 1, we have proved that W(x) generated by Q(z) is also a refinable function vector, and has the same approximation order as U(x). Next we only need prove that W(x) is also an orthogonal. In fact, QðzÞQ ðzÞ þ QðzÞQ ðzÞ ¼ Mðz2 ÞP ðzÞP ðzÞM ðz2 Þ þ Mðz2 ÞP ðzÞP ðzÞM ðz2 Þ ¼ Mðz2 Þ½P ðzÞP ðzÞ þ P ðzÞP ðzÞM ðz2 Þ ¼ Mðz2 ÞM ðz2 Þ ¼ I r : This imply that W(x) generated by Q(z) is also an orthogonal refinable function vector. Now recall that a Hurwitz P polynomial is a polynomial whose zeros all lie in the open left half-plane fz : Re z < 0g. Let pðzÞ ¼ 12 nk¼0 pk zk be a Hurwitz polynomial of exact degree n P 2 which satisfies p(1) = 0 and p(1) = 1. Then, by Theorem 4.1 of [19], there is a unique continuous function /(x) satisfying the following refinement equation: n X pk /ð2x kÞ: ð21Þ /ðxÞ ¼ k¼0
Applying GTST, we extended the above result to the multi case, and obtain the following result. Proposition 1. Let pðzÞ ¼ 12 pð1Þ ¼ 0;
and
Pn
k k¼0 p k z
h
be a Hurwitz polynomial of exact degree n P 2 which satisfies
pð1Þ ¼ 1:
ð22Þ
Assume s · r matrix M(z), and r · s matrix N(z) satisfies N(z)M(z) = Ir. Define QðzÞ ¼ Mðz2 ÞpðzÞN ðzÞ ¼ P k k2Z Qk z : Then there is a unique continuous refinable function vector W(x) satisfying the following refinement equation: X WðxÞ ¼ Qk Wð2x kÞ: ð23Þ 1 2
k2Z
3. Symmetry In this section, we prove that GTST can preserve the symmetry of the given a refinable function vector. Definition 3. U(x) is said to be symmetric, if there is a diagonal matrix EðzÞ ¼ diag ðz2T 1 ; z2T 2 ; . . . ; z2T r Þ; b b such that UðwÞ ¼ EðzÞ UðwÞ. Here, Tj is the point of symmetry of /j(x).
ð24Þ
C. Xie, S. Yang / Applied Mathematics and Computation 194 (2007) 425–430
429
b b Lemma 1 [see [12]]. Suppose that U(x) is symmetric and refinable, Uð2wÞ ¼ P ðzÞ UðwÞ. Then 1
P ðzÞ ¼ Eðz2 ÞP ðzÞEðzÞ ;
ð25Þ
where E(z) is defined by (24). Theorem 4. Let U(x) be symmetric and refinable function vector with two-scale symbol P(z). And suppose that P(z) satisfies (25) with some diagonal matrix E(z). Let Q(z) be the GTST of P(z) with transform matrices M(z) and N(z). That is, QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ. If there exists a diagonal matrix F ðwÞ ¼ diag ðz2t1 ; z2t2 ; . . . ; z2ts Þ, such that ( Mðz2 ÞEðz2 Þ ¼ F ðz2 ÞMðz2 Þ; ð26Þ EðzÞ1 N ðzÞ ¼ N ðzÞF ðzÞ1 : then W(x) generated by Q(z) is also symmetric refinable function vector with multiplicity s. Proof. By Theorem 1, W(x) generated by Q(z) is also a refinable function vector with multiplicity s. Next we prove that W(x) is also symmetric refinable function vector. By (26), 1
1
Mðz2 ÞEðz2 ÞP ðzÞEðzÞ N ðzÞ ¼ F ðz2 ÞMðz2 ÞP ðzÞN ðzÞF ðzÞ : By definition of GTST and symmetry of U(x), we have 1
QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ ¼ F ðz2 ÞQðzÞF ðzÞ : This implies that W(x) is also symmetric refinable function vector. That is, GTST can preserve the symmetry of the given a refinable function vector. h 4. Examples In this section, we will illustrate by some examples how to construct refinable function vector based on GTST. Example 1. Let /D n ðxÞ be Daubechies scaling function (see [2]). p(z) is its two-scale matrix symbol. Choose T 1 3 3 1 3 3 1 þ z þ z2 þ z3 ; 8z þ 9z2 þ 3z3 ; N ðzÞ ¼ 4 þ z2 ; z2 : MðzÞ ¼ 4 4 4 4 2 8 8 It is easy to verify that N(z)M(z) = 1. Let Q(z) of size 2 · 2 be a regular GTST of p(z) with transform matrices M(z) and N(z). That is, pðzÞ 32 þ 108z2 þ 132z4 þ 68z6 þ 12z8 3 10z2 12z4 6z6 z8 2 : QðzÞ ¼ Mðz ÞpðzÞN ðzÞ ¼ 32 1024z2 þ 1536z4 þ 816z6 þ 144z8 96z2 140z4 72z6 12z8 T
Then by Theorem 1, Q(z) can generate an new refinable function vector WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ . 2
Example 2. Let /(x) be a quadratic spline refinable function (see [2]). pðzÞ ¼ ð1þz Þ is its two-scale matrix sym2 T bol. As you know, /ðxÞ is symmetric about the point 1. Choose MðzÞ ¼ ½1; 1 ; N ðzÞ ¼ ½a þ bz; ð1 aÞ bz, for any a, b 2 R. It is easy to verify that M(z) and N(z) satisfy N(z)M(z) = 1 and ( Mðz2 Þz2 ¼ diag ðz2 ; z2 ÞMðz2 Þ; z2 N ðzÞ ¼ N ðzÞdiag ðz2 ; z2 Þ: Hence, by Theorem 4, QðzÞ ¼ Mðz2 Þð1þz Þ2 N ðzÞ can generate an new symmetry refinable function vector 2 T WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ .
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C. Xie, S. Yang / Applied Mathematics and Computation 194 (2007) 425–430
Qm ðzþa Þ n Qmk¼1 k ; where ak > 0; n P 1 and m P 1. It is obvious that p(z) is a Hurwitz Example 3. Let pðzÞ ¼ ð1þz Þ 2 k¼1
ð1þak Þ
polynomial. Hence by Proposition 1 for any s · r matrix M(z) and r · s matrix N(z) which satisfies N(z)M(z) = 1, QðzÞ ¼ Mðz2 ÞpðzÞN ðzÞ can generate a continuous refinable function vector W(x) with multiplicity s. Acknowledgements Supported by the Natural Science Foundation of Guangdong Province (Nos. 06105648, 05008289, and 032038), and the Doctoral Foundation of Guangdong Province (No. 04300917). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
I. Daubechies, Ten Lectures on Wavelets CBM-SNSF Serics in Applied Math # 61, SIAM Publ., Philadelphia, 1992. I. Daubechies, Orthonormal basis of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996. C.K. Chui, J.Z. Wang, On compactly supported spline wavelets and a duality principle, Trans. Am. Math. Soc. 330 (1992) 903–915. T.N.T. Goodman, S.L. Lee, W.S. Tang, Wavelets in wandering subspaces, Trans. Am. Math. Soc. 338 (1993) 639–654. C.K. Chui, J.Z. Wang, A cardinal spline approach to wavelets, Proc. Am. Math. Soc. 113 (1991) 785–793. G. Strang, V. Strela, Orthogonal multiwavelets with vanishing moments, J. Optical Eng. 33 (1994) 2104–2107. J. Geronimo, D.P. Hardin, P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1998) 373–401. G.C. Donovan, J. Geronimo, D.P. Hardin, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996) 1158–1192. G.C. Donovan, J. Geronimo, D.P. Hardin, Intertwining multiresolution analysis and construction of piecewise polynomial wavelets, SIAM J. Math. Anal. 27 (1996) 1791–1815. Yang Shouzhi, Cheng Zhengxing, Wang Hongyong, Construction of biorthogonal multiwavelets, J. Math. Anal. Appl. 276 (2002) 1– 12. Yang Shouzhi, A fast algorithm for constructing orthogonal multiwavelets, ANZIAM J. 46 (2004) 185–202. V. Strela, Multiwavelets: regularity, orthogonality, and symmetry via two-scale similarity transform, Stud. Appl. Math. 98 (1997) 335–354. I. Daubechies, J.C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991) 1388–1410. I. Daubechies, J.C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992) 1031–1079. Q.T. Jiang, Z.W. Shen, On existence and weak stability of matrix refinable functions, Constr. Approx. 15 (1999) 337–353. Z.W. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235–250. C.K. Chui, J.A. Lian, A study on orthonormal multiwavelets, J. Appl. Numer. Math. 20 (1996) 273–298. J.A. Lian, Orthogonal criteria for multiscaling functions, Appl. Comp. Harm. Anal. 5 (1998) 277–311. T.N.T. Goodman, C.A. Micchelli, On refinement equations determined by Po´lya frequency sequences, SIAM J. Math. Anal. 23 (1992) 766–784.
Applied Mathematics and Computation 194 (2007) 425–430 www.elsevier.com/locate/amc
Construction of refinable function vector via GTST Xie Changzhen *, Yang Shouzhi Department of Mathematics, Shantou University, Shantou 515063, PR China
Abstract The concept of general two-scale similarity transform (GTST) is introduced. By applying GTST, we present an algorithm of constructing a class of compactly supported refinable function vector. In addition, we prove that GTST can preserve the symmetry of the given a refinable function vector, and give a sufficient condition that the corresponding transform matrices M(z) and N(z) satisfy when GTST can preserve symmetry. Some examples are given. 2007 Published by Elsevier Inc. Keywords: GTST; Refinable function vector; Symmetry
1. Introduction Several types of uniwavelets are constructed based on multiresolution analysis, such as Daubechies’ orthogonal wavelets [1,2] and semi-orthogonal spline wavelets by Chui and Wang [3]. However, multiwavelets can have some features that uniwavelets cannot. Thus, multiwavelets provide interesting applications in signal processing and some other fields. In recent years, multiscaling functions and multiwavelets have been studied extensively [4–13]. Goodman et al. [4] established a characterization of multiscaling functions and their corresponding multiwavelets. Chui and Wang [5] introduced semi-orthogonal spline multiwavelets. Examples of cubic and quintic finite elements and their corresponding multiwavelets were studied by Strang and Strela [6]. Geronimo et al. [7] used fractal interpolation to construct orthogonal multiscaling functions, and their corresponding multiwavelets were given in [8]. In [9], Donovan, Geronimo, and Hardin showed that there exist compactly supported orthogonal polynomial spline multiscaling functions with arbitrarily high regularity. Are there any easier methods to construct refinable function vector? Can refinable function vector be constructed based on scalar refinable function? As you know, the two-scale similarity transform (TST) is a new, nonobvious construction for multiwavelets [12]. Regular TST have some applications in verifying and imposing symmetry conditions. In this paper, by means of idea of Strela [12], we introduce the concept of GTST, and apply GTST to construct a class of compactly supported refinable function vector.
*
Corresponding author. E-mail address: [email protected] (C. Xie).
0096-3003/$ - see front matter 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2006.12.088
426
C. Xie, S. Yang / Applied Mathematics and Computation 194 (2007) 425–430
2. GTST T
Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be a refinable function vector, satisfying the following two-scale matrix equation X UðxÞ ¼ P k Uð2x kÞ; ð1Þ k2Z
where some r · r matrices {Pk} called the two-scale matrix sequence. If r = 1, {Pk} are real number sequence and U(x) = /1(x). In this case we say that (1) is a scalar dilation refinement equation. Solutions of Eq. (1) have been extensively studied in recent years [13–18]. It turned out that it is more conb venient to work with the Fourier transform UðwÞ that with the refinable function U(x) itself. By taking Fourier transform on both sides of (1), we have b b w ; z ¼ eiw=2 ; UðwÞ ¼ P ðzÞ U ð2Þ 2 P where P ðzÞ ¼ 12 k2Z P k zk called two-scale matrix symbol corresponding to U(x). In the scalar case P(z) is a trigonometric polynomial. In the multi case P(z) becomes a matrix trigonometric polynomial. We called that U(x) generates a space V0 if V0 consists of all finitely linear combination of integer translates of entries of U(x). Assume that U(x) generates a space V0. Then for any compactly supported functions w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ in V0, there exists a finitely many nonzero matrices Mk of size s · r such that X T WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ ¼ M k Uðx kÞ: ð3Þ k2Z
In terms of Fourier transform, we have b b WðwÞ ¼ MðzÞ UðwÞ;
P
ð4Þ k
where M(z) denotes the s · r matrix of Laurent polynomials, i.e., MðzÞ ¼ k2Z M k z . Specially, in the case of r = s, M(z) is a square matrix. A square matrix M(z) is said to be invertible if detðMðzÞÞ is a monomial of z. That is, detðMðzÞÞ ¼ kzm for a scalar k 5 0 and an integer m 2 Z. It is clear that if M(z) is invertible, W(x) generates the same space S. In this case, we have 1 b b b b Wð2wÞ ¼ Mðz2 Þ Uð2wÞ ¼ Mðz2 ÞP ðzÞ UðwÞ ¼ Mðz2 ÞP ðzÞMðzÞ WðwÞ:
ð5Þ
This means that W(x) is an new refinable function vector with two-scale matrix symbol 1
QðzÞ ¼ Mðz2 ÞP ðzÞMðzÞ :
ð6Þ
The case of r = s is well studied by Strela [12]. Definition 1. Assume M(z) of size r · r is a trigonometric matrix polynomial which is invertible. Q(z) is a regular TST of P(z) if QðzÞ ¼ Mðz2 ÞP ðzÞMðzÞ1 :
ð7Þ
e QðzÞ is a regular inverse TST(ITST) of P(z) if e QðzÞ ¼ Mðz2 Þ1 P ðzÞMðzÞ:
ð8Þ
Regular TST have some applications in verifying and imposing symmetry conditions. See [12] for details. By means of idea of Strela [12], we present the concept of GTST. Definition 2. Assume M(z) of size s · r is a trigonometric matrix polynomial, and N(z) of size r · s is a trigonometric matrix polynomial which satisfy N(z)M(z) = Ir. Q(z) of size s · s is a regular GTST of P(z) if QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ:
ð9Þ
C. Xie, S. Yang / Applied Mathematics and Computation 194 (2007) 425–430
427
T
Theorem 1. Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be a refinable function vector with two-scale matrix symbol P(z) of size r · r. Assume M(z) of size s · r is a trigonometric matrix polynomial, and N(z) of size r · s is a trigonometric matrix polynomial which satisfy N(z)M(z) = Ir. Let Q(z) of size s · s be a regular GTST of P(z) with T transform matrices M(z) and N(z). Then WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞ generated by Q(z) is an new refinable function vector with multiplicity s, and has the same approximation order as U(x). Proof. Construct function vector WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; ws ðxÞT as follows: X M k Uðx kÞ: WðxÞ ¼
ð10Þ
k2Z
By taking Fourier transform the both sides of (10), we have b b WðwÞ ¼ MðzÞ UðwÞ:
ð11Þ
Hence b b b b ¼ Mðz2 ÞP ðzÞ UðwÞ ¼ Mðz2 ÞP ðzÞN ðzÞ WðwÞ: Wð2wÞ ¼ Mðz2 Þ Uð2wÞ
ð12Þ
This means that W(x) is also refinable function with multiplicity s. Its two-scale matrix symbol QðzÞ ¼ b b Mðz2 ÞP ðzÞN ðzÞ. By (11), it is clear that UðwÞ ¼ N ðzÞ WðwÞ. That is, X UðxÞ ¼ N ‘ Wðx ‘Þ: ð13Þ ‘2Z
Suppose that U(x) has approximation order p P 1. Hence, there exists a set of row vectors fujk g R1r that satisfy X j uk Uðx kÞ; j ¼ 0; 1; . . . ; p 1: xj ¼ k2Z
By (13), we have XX j xj ¼ uk N ‘k Wðx ‘Þ; j ¼ 0; 1; . . . ; p 1: k2Z
‘2Z
The above implies that W(x) has also approximation order p. This completes the proof of Theorem 1. Let /(x) be a scalar refinable function. Hence, there exists sequence {pk} such that X pk /ð2x kÞ: ð14Þ /ðxÞ ¼ k2Z
where pðzÞ ¼ 12 Define A0 ðzÞ ¼
P
k2Z p k z
k
1X p zk ; 2 k2Z 2k
called two-scale sequence symbol corresponding to /(x).
A1 ðzÞ ¼
1X p zk : 2 k2Z 2kþ1
ð15Þ
It follows that pðzÞ ¼ A0 ðz2 Þ þ zA1 ðz2 Þ:
ð16Þ
Theorem 2. There exist two Laurent polynomials B0(z) and B1(z) of degree 6n such that A0 ðzÞB0 ðzÞ þ A1 ðzÞB1 ðzÞ ¼ 1:
ð17Þ
Proof. Recalling (16), we see that 1 1 A0 ðz2 Þ ¼ ½pðzÞ þ pðzÞ; zA1 ðz2 Þ ¼ ½pðzÞ pðzÞ: 2 2
ð18Þ
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Since p(z) is a two-scale sequence symbol, p(z) and p(z) have no common zeros (see [1]). Hence A0(z) and A1(z) have no common zeros. By applying Bezout Theorem, there exist two Laurent polynomials B0(z) and B1(z) of degree 6n such that B0(z) and B1(z) satisfies (17). This completes the proof of our claim. Applying the above A0(z), A1(z), B0(z) and, B1(z), we can construct M(z), N(z) as follows: ( T Mðz2 Þ ¼ ½A0 ðz2 Þ þ zA1 ðz2 Þ; B1 ðz2 Þ þ zB0 ðz2 Þ ; ð19Þ N ðz2 Þ ¼ ½B0 ðz2 Þ; A1 ðz2 Þ: It is easy to verify that N ðzÞMðzÞ ¼ 1:
ð20Þ
Next, we discuss the case of r = s. If r = s, then GTST becomes TST. Specially, if N ðzÞMðzÞ ¼ I r and N ðzÞ ¼ MðzÞ , then N(z)(or M(z)) is a paraunitary matrix. h T
Theorem 3. Let UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /r ðxÞ be an orthogonal refinable function vector with two-scale matrix symbol P(z) of size r · r. Assume r · r matrix M(z) is a paraunitary matrix. Let Q(z) of size r · r be a regular GTST of P(z) with transform matrices M(z) and MðzÞ . Then WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ; . . . ; wr ðxÞT generated by Q(z) is also an orthogonal refinable function vector, and has the same approximation order as U(x). Proof. In Theorem 1, we have proved that W(x) generated by Q(z) is also a refinable function vector, and has the same approximation order as U(x). Next we only need prove that W(x) is also an orthogonal. In fact, QðzÞQ ðzÞ þ QðzÞQ ðzÞ ¼ Mðz2 ÞP ðzÞP ðzÞM ðz2 Þ þ Mðz2 ÞP ðzÞP ðzÞM ðz2 Þ ¼ Mðz2 Þ½P ðzÞP ðzÞ þ P ðzÞP ðzÞM ðz2 Þ ¼ Mðz2 ÞM ðz2 Þ ¼ I r : This imply that W(x) generated by Q(z) is also an orthogonal refinable function vector. Now recall that a Hurwitz P polynomial is a polynomial whose zeros all lie in the open left half-plane fz : Re z < 0g. Let pðzÞ ¼ 12 nk¼0 pk zk be a Hurwitz polynomial of exact degree n P 2 which satisfies p(1) = 0 and p(1) = 1. Then, by Theorem 4.1 of [19], there is a unique continuous function /(x) satisfying the following refinement equation: n X pk /ð2x kÞ: ð21Þ /ðxÞ ¼ k¼0
Applying GTST, we extended the above result to the multi case, and obtain the following result. Proposition 1. Let pðzÞ ¼ 12 pð1Þ ¼ 0;
and
Pn
k k¼0 p k z
h
be a Hurwitz polynomial of exact degree n P 2 which satisfies
pð1Þ ¼ 1:
ð22Þ
Assume s · r matrix M(z), and r · s matrix N(z) satisfies N(z)M(z) = Ir. Define QðzÞ ¼ Mðz2 ÞpðzÞN ðzÞ ¼ P k k2Z Qk z : Then there is a unique continuous refinable function vector W(x) satisfying the following refinement equation: X WðxÞ ¼ Qk Wð2x kÞ: ð23Þ 1 2
k2Z
3. Symmetry In this section, we prove that GTST can preserve the symmetry of the given a refinable function vector. Definition 3. U(x) is said to be symmetric, if there is a diagonal matrix EðzÞ ¼ diag ðz2T 1 ; z2T 2 ; . . . ; z2T r Þ; b b such that UðwÞ ¼ EðzÞ UðwÞ. Here, Tj is the point of symmetry of /j(x).
ð24Þ
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429
b b Lemma 1 [see [12]]. Suppose that U(x) is symmetric and refinable, Uð2wÞ ¼ P ðzÞ UðwÞ. Then 1
P ðzÞ ¼ Eðz2 ÞP ðzÞEðzÞ ;
ð25Þ
where E(z) is defined by (24). Theorem 4. Let U(x) be symmetric and refinable function vector with two-scale symbol P(z). And suppose that P(z) satisfies (25) with some diagonal matrix E(z). Let Q(z) be the GTST of P(z) with transform matrices M(z) and N(z). That is, QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ. If there exists a diagonal matrix F ðwÞ ¼ diag ðz2t1 ; z2t2 ; . . . ; z2ts Þ, such that ( Mðz2 ÞEðz2 Þ ¼ F ðz2 ÞMðz2 Þ; ð26Þ EðzÞ1 N ðzÞ ¼ N ðzÞF ðzÞ1 : then W(x) generated by Q(z) is also symmetric refinable function vector with multiplicity s. Proof. By Theorem 1, W(x) generated by Q(z) is also a refinable function vector with multiplicity s. Next we prove that W(x) is also symmetric refinable function vector. By (26), 1
1
Mðz2 ÞEðz2 ÞP ðzÞEðzÞ N ðzÞ ¼ F ðz2 ÞMðz2 ÞP ðzÞN ðzÞF ðzÞ : By definition of GTST and symmetry of U(x), we have 1
QðzÞ ¼ Mðz2 ÞP ðzÞN ðzÞ ¼ F ðz2 ÞQðzÞF ðzÞ : This implies that W(x) is also symmetric refinable function vector. That is, GTST can preserve the symmetry of the given a refinable function vector. h 4. Examples In this section, we will illustrate by some examples how to construct refinable function vector based on GTST. Example 1. Let /D n ðxÞ be Daubechies scaling function (see [2]). p(z) is its two-scale matrix symbol. Choose T 1 3 3 1 3 3 1 þ z þ z2 þ z3 ; 8z þ 9z2 þ 3z3 ; N ðzÞ ¼ 4 þ z2 ; z2 : MðzÞ ¼ 4 4 4 4 2 8 8 It is easy to verify that N(z)M(z) = 1. Let Q(z) of size 2 · 2 be a regular GTST of p(z) with transform matrices M(z) and N(z). That is, pðzÞ 32 þ 108z2 þ 132z4 þ 68z6 þ 12z8 3 10z2 12z4 6z6 z8 2 : QðzÞ ¼ Mðz ÞpðzÞN ðzÞ ¼ 32 1024z2 þ 1536z4 þ 816z6 þ 144z8 96z2 140z4 72z6 12z8 T
Then by Theorem 1, Q(z) can generate an new refinable function vector WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ . 2
Example 2. Let /(x) be a quadratic spline refinable function (see [2]). pðzÞ ¼ ð1þz Þ is its two-scale matrix sym2 T bol. As you know, /ðxÞ is symmetric about the point 1. Choose MðzÞ ¼ ½1; 1 ; N ðzÞ ¼ ½a þ bz; ð1 aÞ bz, for any a, b 2 R. It is easy to verify that M(z) and N(z) satisfy N(z)M(z) = 1 and ( Mðz2 Þz2 ¼ diag ðz2 ; z2 ÞMðz2 Þ; z2 N ðzÞ ¼ N ðzÞdiag ðz2 ; z2 Þ: Hence, by Theorem 4, QðzÞ ¼ Mðz2 Þð1þz Þ2 N ðzÞ can generate an new symmetry refinable function vector 2 T WðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ .
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Qm ðzþa Þ n Qmk¼1 k ; where ak > 0; n P 1 and m P 1. It is obvious that p(z) is a Hurwitz Example 3. Let pðzÞ ¼ ð1þz Þ 2 k¼1
ð1þak Þ
polynomial. Hence by Proposition 1 for any s · r matrix M(z) and r · s matrix N(z) which satisfies N(z)M(z) = 1, QðzÞ ¼ Mðz2 ÞpðzÞN ðzÞ can generate a continuous refinable function vector W(x) with multiplicity s. Acknowledgements Supported by the Natural Science Foundation of Guangdong Province (Nos. 06105648, 05008289, and 032038), and the Doctoral Foundation of Guangdong Province (No. 04300917). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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