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c 1988 Society for Industrial and Applied Mathematics

SIAM J. MATH. ANAL. Vol. 1, No. 2, pp. 1{XX, February 1988

001

ANALYSIS AND CONSTRUCTION OF OPTIMAL MULTIVARIATE BIORTHOGONAL WAVELETS WITH COMPACT SUPPORT BIN HANy

Abstract. In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1  p  1) smoothness of multivariate re nable functions is presented. It is well known that there is a close relation between a fundamental re nable function and a biorthogonal wavelet. We shall demonstrate that any fundamental re nable function, whose mask is supported on [1 ? 2r; 2r ? 1]s for some positive integer r and satis es the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental re nable function with the mask br . Here the sequence br on Z is the unique univariate interpolatory re nement mask which is supported on [1 ? 2r; 2r ? 1] and satis es the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal scaling function, whose mask is supported on [0; 2r ? 1]s for some positive integer r and satis es the sum rules of optimal order r, has Lp smoothness not exceeding that of the univariate Daubechies orthogonal scaling function whose mask is supported on [0; 2r ? 1]. We also demonstrate that a similar result holds true for biorthogonal wavelets. Examples are provided to illustrate the general theory. Finally, a general CBC (Construction By Cosets) algorithm is presented to construct all the dual re nement masks of any given interpolatory re nement mask with the dual masks satisfying arbitrary order of sum rules. Thus, for any scaling function which is fundamental, this algorithm can be employed to generate a dual scaling function with arbitrary approximation order. This CBC algorithm can be easily implemented. As a particular application of the general CBC algorithm, a TCBC (Triangle Construction By Cosets) algorithm is proposed. For any positive integer k and any interpolatory re nement mask a such that a is symmetric about all the coordinate axes, such TCBC algorithm provides us a dual mask of a such that the dual mask satis es the sum rules of order 2k and is also symmetric about all the coordinate axes. As an application of this TCBC algorithm, a family of optimal bivariate biorthogonal wavelets is presented with the scaling function being a spline function. Key words. Biorthogonal wavelets, orthogonal wavelets, interpolatory subdivision schemes, fundamental functions, sum rules, Lp smoothness, critical exponent, algorithm AMS subject classi cations. 65D05, 41A25, 46E35, 41A05, 41A63, 41A30

1. Introduction. Based on the work [25, 26], the present paper deals with the analysis and construction of multivariate biorthogonal wavelets with some desired properties. It is well known that in various applications high smoothness, small support and high vanishing moments are the three most important properties of a (bi)orthogonal wavelet. On the other hand, there is no C 1 (bi)orthogonal wavelet with compact support. In this paper, we shall investigate the mutual relations among these three properties. Compactly supported (bi)orthogonal wavelets on the real line have been found to be very useful in applications such as signal processing and image compression, for examples, see [1, 16, 36, 37]. In [10], Cohen, Daubechies and Feauveau proposed a general way of constructing univariate biorthogonal wavelets. Though the tensor product (bi)orthogonal wavelets provide a family of multivariate (bi)orthogonal wavelets  Received

the editors March 31, 1998; accepted by the editors February 19, 1999. y Program inby Applied and Computational Mathematics, Princeton University, Princeton, NJ

08544, USA ([email protected], http://www.math.princeton.edu/bhan). The research of this author was supported by Killam Trust under Isaak Walton Killam Memorial Scholarship. Permanent address: ([email protected], http://xihu.math.ualberta.ca/bhan). 1

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to deal with problems in high dimensions in applications, it has its own advantages and disadvantages. Therefore, as noted in many papers [5, 9, 12, 24, 27, 37, 42] and references cited there, it is of interest in its own right to construct non-tensor product (bi)orthogonal wavelets in the high dimensions. In the current literature, there are many papers on constructing multivariate biorthogonal wavelets, especially bivariate biorthogonal wavelets. To mention only a few here, see [9, 12, 24, 27, 37, 42] and references therein. Bivariate compactly supported quincunx biorthogonal wavelets were constructed by Cohen and Daubechies in [9]. In [42], a family of bivariate biorthogonal wavelets with the scaling function being a box spline was given by Riemenschneider and Shen. Usually, a biorthogonal wavelet is derived from a multiresolution analysis generated by a pair of a scaling function and its dual scaling function. The construction of wavelets in the multivariate setting is more challenging than its univariate counterpart, see [3, 10, 16, 24, 27, 33, 35, 38, 42] and references therein on construction of (bi)orthogonal wavelets from a multiresolution analysis. To obtain a biorthogonal wavelet, we have to nd two re nable functions with some desired properties. A function  is said to be re nable if it satis es the following re nement equation (1:1)

=

X

2Zs

a( )(2
? );

where a P is a nitely supported sequence on Zs, called the re nement mask. If a satis es 2Zs a( ) = 2s ; then it is known (see [4]) that there exists a unique compactly supported distribution  satisfying the re nement equation (1.1) subject to the condition b(0) = 1. This distribution is said to be the normalized solution of the re nement equation (1.1). Throughout this paper we shall use a to denote the normalized solution of the re nement equation (1.1) with the mask a. The concepts of linear independence and approximation order of a function play an important role in the study of biorthogonal wavelets. The shifts of a compactly supported function  : Rs ! C are said to be linearly independent if for any z 2 C s , there exists a multi-integer in Zs such that b(z + 2 ) 6= 0. If for any  2 Rs , there exists a multi-integer in Zs such that b( + 2 ) 6= 0, then the shifts of  are said to be stable. See [34] for discussion on linear independence and stability. By `(Zs) we denote the linear space of all sequences on Zs. For a compactly supported function  in Lp(Rs ) (1  p  1), we de ne

S () :=

X

2Zs

(
? )b() : b 2 `(Zs)



and call it the shift-invariant space generated by . For h > 0, the scaled space S h is de ned by S h := ff (
=h) : f 2 S ()g. For a positive integer k, we say that S () provides approximation order k if for each suciently smooth function f in Lp (Rs ), there exists a positive constant C such that infh kf ? gkp  Chk 8 h > 0: g2S

The general procedure of constructing a biorthogonal wavelet is the following. First, nd a re nable function  in L2 (Rs ) such that  satis es the re nement equation (1.1) with a nitely supported re nement mask a and the shifts of  are linearly independent. Such function  is called a scaling function. The next step is to nd a

optimal multivariate biorthogonal wavelets

3

re nable function d in L2 (Rs ) such that d satis es

d =

(1:2)

X

2Z

s

ad( )d (2
? );

where ad is a nitely supported sequence on Zs, and d satis es the following biorthogonal relation

Z

(1:3)

Rs

(t ? )d (t) dt = ()

8  2 Zs;

where (0) = 1 and () = 0 for all  2 Zsnf0g. This function d is called a dual scaling function of . If  is the dual scaling function of itself,  is called an orthogonal scaling function. Finally, a biorthogonal wavelet is derived from the above ; d ; a and ad. The reader is referred to [5, 6, 7, 10, 16, 24, 27, 33, 35, 38, 42] for detail on the construction of a biorthogonal wavelet from a pair of a scaling function and its dual scaling function. It is well known that the smoothness of the scaling function and its dual scaling function will determine the smoothness of their derived wavelets, and the approximation orders of the scaling function and its dual scaling function will determine the vanishing moments of their derived wavelets. For more detail on (bi)orthogonal wavelets, the reader is referred to [3, 5, 6, 7, 9, 10, 12, 13, 14, 16, 24, 27, 33, 35, 37, 42, 44] and references cited there. By we denote the set of the vertices of the unit cube [0; 1]s . For a positive integer k, we say that a sequence a on Zs satis es the sum rules of order k if X X (1:4) a(2 + ")p(2 + ") = a(2 )p(2 ) 8 " 2 ; p 2 k?1 ; 2Zs

2Zs

where k?1 is the set of polynomials with total degree less than k. Let a function  be a re nable function with a mask a. It was proved by Jia in [30, 31] that if the shifts of  are stable, then S () provides approximation order k if and only if the mask a satis es the sum rules of order k. Therefore, it is evident that S () (or S (d )) provides approximation order k if and only if the mask a (or ad ) satis es the sum rules of order k.

Now it is natural to ask the following question: given a scaling function with compact support, does a dual scaling function with compact support exist? As noted by Lemarie [39], the answer is yes at least in the univariate case. More precisely, given a scaling function with compact support, a dual scaling function always exists with compact support and arbitrarily high smoothness. Therefore, it is valuable to ask that for a scaling function, if we x the size of the support of a dual scaling function, then what is the highest approximation order and the highest smoothness of a dual scaling function that we can expect? Based on our previous work on interpolatory subdivision schemes [25, 26], we shall answer the above question in this paper. Here is an outline of this paper. In Section 2, given a scaling function, we shall study the relation between the approximation order of its dual scaling function and the support of its dual scaling function. In Section 3, a characterization of Lp smoothness of a multivariate re nable function is given. In Section 4, we shall prove that any orthogonal scaling function, whose mask is supported on [0; 2r ? 1]s (r 2 N ) and satis es the sum rules of optimal order r, has Lp smoothness not exceeding that of the univariate Daubechies orthogonal scaling function whose mask is supported on [0; 2r ? 1]. An example will be provided to illustrate our result. In Section 5, we rst study the optimal Lp smoothness of a fundamental re nable function if its

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mask is supported on [1 ? 2r; 2r ? 1]s and satis es the sum rules of optimal order 2r. Next for any given scaling function, we shall study the optimal smoothness of a dual scaling function if its support is xed and it attains the optimal approximation order. Finally, in Section 6, a general CBC (Construction By Cosets) algorithm is presented to generate all the dual masks of a given interpolatory re nement mask. This algorithm can be easily implemented. In particular, as an application of this general construction, we shall propose a TCBC (Triangle Construction By Cosets) algorithm such that for any bivariate interpolatory mask which is symmetric about the two coordinate axes, we can construct a family of dual masks with arbitrary order of sum rules and symmetry about the two coordinate axes. At the end of this paper, a family of optimal bivariate biorthogonal wavelets is constructed from a spline scaling function. 2. Optimal Approximation Order of a Dual Scaling Function. In this section, we shall rst introduce some notation. For a given scaling function, we shall study the relation between the approximation order of a dual scaling function and the support of a dual scaling function. In order to solve the re nement equation (1.1), we start with an initial function 0 given by

0 (x1 ;


; xs ) :=

Ys

j =1

(xj );

(x1 ;


; xs ) 2 Rs ;

where  is the univariate hat function de ned by (x) := maxf1 ? jxj; 0g; x 2 R: Then we employ the iteration scheme Qna 0 ; n = 0; 1; 2;


; where Qa is the bounded linear operator on Lp (Rs ) (1  p  1) given by (2:1)

Qa f :=

X

2Zs

a( )f (2
? );

f 2 Lp (Rs ):

This iteration scheme is called a subdivision scheme or a cascade algorithm associated with the mask a (see [4, 17]). For any p such that 1  p  1, we say that the subdivision scheme associated with a mask a converges in the Lp norm if there exists a function f in Lp (Rs ) such that limn!1 kQna 0 ? f kp = 0. If this is the case, then the limit function f must be the normalized solution of the re nement equation (1.1) with the re nement mask a. Before proceeding further, we introduce some notation. By `(Zs) we denote the space of all sequences on Zs, and by `0 (Zs) the linear space of all nitely supported sequences on Zs. By  we denote the element given by (0) = 1 and ( ) = 0 for all 2 Zsnf0g. For j = 1;


; s; let ej be the j th coordinate unit vector. The dierence operator rj on `(Zs) is de ned by rj  :=  ? (
? ej );  2 `(Zs). The subdivision operator associated with a mask a is de ned by (2:2)

Sa () :=

X

2Zs

a( ? 2 )( );

 2 Zs;

where  2 `0 (Zs). It was proved in [25] that the subdivision scheme associated with a mask a converges in the Lp norm if and only if  lim max krj San k1p=n : j = 1;


; s < 2s=p : n!1

optimal multivariate biorthogonal wavelets

5

It is well known that there is a close relation between biorthogonal wavelets and fundamental re nable functions. A function  is said to be fundamental if  is continuous, (0) = 1, and () = 0 for all  2 Zsnf0g. If  is a fundamental re nable function with a mask a, then it is necessary that a(0) = 1 and a(2 ) = 0 8 2 Zsnf0g: A mask that satis es the above condition is called an interpolatory re nement mask. The following fact is well known (see [8, 16, 40]) and reveals the relation between a biorthogonal wavelet and a fundamental re nable function. Lemma 2.1. Let a function  be a scaling function with a mask a, and let d be a dual scaling function of  with a mask ad . De ne

Z

(2:3)

(x) :=

and (2:4)

b() := 2?s

Rs

(t ? x)d (t) dt;

X 2Zs

x 2 Rs ;

a( ? )ad( );

 2 Zs:

Then the function  is a fundamental re nable function satisfying the re nement equation (1:1) with the interpolatory mask b. In other words, the mask a and ad satisfy the following well-known discrete biorthogonal relation:

X

(2:5)

2Zs

a( ? 2) ad( ) = 2s ()

8  2 Zs:

Conversely, if the masks a and ad satisfy the above discrete biorthogonal relation (2:5) and the subdivision schemes associated with a and ad converge in the L2 norm respectively, then the functions  and d lie in L2 (Rs ) and satisfy the biorthogonal relation (1:3) where the functions  and d are the normalized solutions of the re nement equations (1:1) with the masks a and ad respectively. Therefore, the function  is a scaling function and d is a dual scaling function of . If two sequences a and ad on Zs satisfy the discrete biorthogonal relation (2.5), then the mask ad is called a dual mask of the mask a. Throughout this paper, we shall use the following notation:

:= f (z1;


; zs ) 2 C s : jz1 j =


= jzs j = 1 g: For any sequence  in `0 (Zs), its symbol e is given by Ts

(2:6)

e(z ) :=

X

2Z

s

( )z ;

z 2 Ts :

By Lemma 2.1, we have b ( ) = b( )cd ( );  2 Rs and eb(z ) = 2?sea(z )aed (z ); z 2 Ts. The following result was proved in [26] and will be needed later. Theorem 2.2. (see [26, Theorem 2.1 and Theorem 2.2]) Suppose that a is an interpolatory mask supported on Zs \ sj=1 [?Lj ; Hj ] for some nonnegative integers Lj and Hj . If the mask a satis es the sum rules of order k, then  L + 1   Hj + 1  j + ; k  1min js 2 2

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where b
c is the oor function. Moreover, when s = 1, there exists a unique interpolatory re nement mask supported on [?L1; H1 ] and satisfying the sum rules of order b L12+1 c + b H12+1 c. By the above theorem, in the univariate case (s = 1), there is a unique interpolatory mask supported on [1 ? 2r; 2r ? 1] and satisfying the sum rules of order 2r. This is the same interpolatory mask as given by Deslauriers and Dubuc in [18], and will be denoted by br throughout this paper. In the multivariate case (s > 1), such interpolatory masks are not unique. Let tr be the sequence on Zs given by

tr (1 ; : : : ; s ) := br (1 )


br (s );

(2:7)

(1 ; : : : ; s ) 2 Zs:

Then tr is a tensor product interpolatory re nement mask supported on [1 ? 2r; 2r ? 1]s and it satis es the sum rules of the optimal order 2r. Based on the above results, we have the following theorem: Theorem 2.3. Let  be a scaling function with its re nement mask a supported on sj=1 [?lj ; hj ] for some nonnegative integers lj and hj , and d be its dual scaling function with its mask ad supported on sj=1 [?Lj ; Hj ] for some nonnegative integers Lj and Hj . Suppose that a satis es the sum rules of order k, then ad can satisfy the sum rules of order at most min

1j s

 hj + Lj + 1   lj + Hj + 1  + ? k; 2

2

where b
c is the oor function. Proof. Let b be the sequence de ned in (2.4). Then by Lemma 2.1, b is an interpolatory mask and b is supported on sj=1 [?hj ? Lj ; lj + Hj ]. From Theorem ? h j +Lj +1 2.2, we see
that b can satisfy the sum rules of order at most min1js b 2 c + b lj +H2 j +1 c . To complete the proof, it suces to prove that if the mask ad satis es the sum rules of order ek, then b will satisfy the sum rules of order at least k + ek. Denote Zs+ :=

f (1 ;


; s ) 2 Zs : j  0 8 j = 1;


; s g;

and jj := 1 +


+ s for  = (1 ;


; s ) 2 Zs+, and  := 1 1


s s for  = (1 ;


; s ) 2 Zs. Thus, by the de nition of the sum rules given in (1.4), it suces to prove that

X

2Zs

b(2 + ")(2 + ") =

X

2Zs

b(2)(2)

8  2 Zs+; jj < k + ek; " 2 :

By the de nition of the sequence b given in (2.4), we can rewrite the above equality as follows: for any " 2 and  2 Zs+ such that jj < k + ek,

XX

2Zs 2Zs

a( ? 2 ? ")ad( )(2 + ") =

XX

2Zs 2Zs

a( ? 2)ad( )(2) :

Therefore, it suces to prove that the left side of the above equality

C" :=

XXX

"0 2 2Zs 2Zs

a(2 + "0 ? 2 ? ")ad (2 + "0 )(2 + ")

optimal multivariate biorthogonal wavelets

does not depend on " for any  2 Zs+ such that jj < k + ek. On the other hand,

?

7




(2 + ") = (2 + "0 ) ? (2 + "0 ? 2 ? ")  X = (?1)j j  !(?!  )! (2 + "0 ? 2 ? ") (2 + "0 )? ; 0 

where ! := 1 !


s ! for  = (1 ;


; s ), and    if and only if j  j for all j = 1;


; s. Thus, C" can be rewritten as X jj ! X X X C" = (?1)  !( ?  )! 0  2Zs "0 2 2Zs

a(2 + "0 ? 2 ? ")(2 + "0 ? 2 ? ") ad (2 + "0 )(2 + "0 )? : Therefore, it suces to demonstrate that for any  2 Zs+ such that 0    , C"; :=

XX

"0 2 2Zs

X

a(2 + "0 ? ")(2 + "0 ? ")

2Zs

ad(2 + "0 )(2 + "0 )?

does not depend on ". Note that j ?P j + j j = jj < k + ek implies that either j ?  j < k or j j < ek. If j j < k, then 2Zs a(2 + "0 ? ")(2 + "0 ? ") does not depend on both " and "0 since the sequence a satis es the sum rules of order k. Hence, for any  in Zs+ such that j j < k, we have

C"; =

X

2Z

s

a(2)(2)

X

2Z

s

ad ( ) ?

P

does not depend on ". Similarly, if j ?  j < ek, then 2Zs ad (2 + "0)(2 + "0 )? does not depend on "0 since the sequence ad satis es the sum rules of order ek. Therefore,

C"; =

X

2Zs

ad (2 )(2 )?

X

2Zs

a()

does not depend on " which completes the proof. From the proof of Theorem 2.3, it is straightforward to obtain the following result: Corollary 2.4. If a function  in L2 (Rs ) is an orthogonal scaling function with its mask a supported on [0; r]s for some positive integer r, then the mask a can satisfy the sum rules of order at most b r+1 2 c. Therefore, S () can provide approximation c . order at most b r+1 2

3. Characterization of Lp Smoothness of a Re nable Function. In this section, we will study the smoothness of a re nable function in the multivariate setting. Many results on the analysis of L2 smoothness of a re nable function both in the univariate case and in the multivariate case are obtained in the current literature. To mention only a few here, see [11, 17, 23, 29, 41, 43, 45] and references therein. For s = 1, the characterization of Lp smoothness was given by Villemoes in [45]. In this section, based on a result of Ditzian [19, 20], we present a simple proof to characterize the Lp smoothness of a multivariate re nable function. Jia will discuss the Lp smoothness of a re nable function with an arbitrary dilation matrix in a forthcoming paper [32].

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We shall use the generalized Lipschitz space to measure smoothness of a given function. For any vector y in Rs , the dierence operator ry on Lp (Rs ) is de ned to be ry f = f ? f (
? y); f 2 Lp (Rs ): Let k be a positive integer. The k-th modulus of smoothness of a function f in Lp(Rs ) is de ned by !k (f; h)p := sup krky f kp; h > 0: jyjh

For ? > 0, let
k be an integer greater than  . The generalized Lipschitz space Lip ; Lp (Rs ) consists of those functions f in Lp (Rs ) for which (3:1) !k (f; h)p  Ch 8 h > 0; where C is a constant independent of h, or in other words, !k (f; h)p = O(h ). The Lp smoothness of a function f 2 Lp (Rs ) in the Lp norm sense is described by its Lp critical exponent p (f ) de ned by  ?
(3:2) p (f ) := sup  : f 2 Lip ; Lp(Rs ) : In the following, we will characterize the Lp (1  p  1) smoothness of a re nable function in multidimensional spaces. To do this, we need the following result on moduli of smoothness, which is based on a result of Ditzian in [19, 20]. Theorem 3.1. Let f be a function in
Lp(Rs ) and  be a positive real number. ?  Then f belongs to the space Lip ; Lp(Rs ) if and only if for an integer k greater than  , there exists a positive constant C such that (3:3) maxf krk2?nei f kp : i = 1;


; s g  C 2?n 8 n 2 N; where ei is the i-th coordinate unit vector. ?
Proof. Necessity:? If f belongs to Lip ; Lp(Rs ) , then by the de nition of the
Lipschitz space Lip ; Lp (Rs ) , there exists a positive constant C such that

krk2?n ei f kp  !k (f; 2?n )  C 2?n

8 1  i  s; n 2 N :

Hence inequality (3.3) holds true. Suciency: If inequality (3.3) holds true, then we can demonstrate that there exists a positive constant C1 such that (3:4) krkhei f kp  C1 h 8 1  i  s; h > 0: Let g be a simple function such that kgkq = 1 where 1=p + 1=q = 1. De ne

F (x) := f  g(x) =

Z

Rs

f (x ? t)g(t) dt;

x 2 Rs :

Then the function F is continuous and bounded. Note that the inequality (3.3) implies that for any i = 1;


; s, krk2?nei F k1 = k(rk2?nei f )  gk1  krk2?nei f kpkgkq  C 2?n 8 n 2 N : Therefore, in particular, we have jrk2?n ei F (tei )j  C 2?n 8 t 2 R; n 2 N :

optimal multivariate biorthogonal wavelets

9

By a result of Boman [2, Theorem 1] and Ditzian [20], there exists a positive constant C1 depending only on k and C (independent of g) such that

jrkhei F (tei )j  C1 h 8 t 2 R; h > 0: Note that rkhei F (0) = (rkhei f )  g(0). It follows from the above inequality (3.5) that for any simple function g with kgkq = 1, we have that for any i = 1;


; s, (3:5)

Z ? rkhe f
(?x)g(x) dx = j(rkhe f )  g(0)j = jrkhe F (0)j  C1h 8 h > 0: R s

This yields

i

i

i

Z ?
rkhe f (?x)g(x) dx  C1 h

krkhei f kp = sup kgkq =1

Rs

i

8 1  i  s; h > 0:

Therefore, inequality (3.4) is veri ed. By inequality (3.4) and a result of Ditzian [19, Corollary 5.2, and also cf. Theorem 5.1], to see that the function ? it is straightforward
f belongs to the function space Lip ; Lp(Rs ) : Remark 1. In fact, the result in Corollary 5.2 of Ditzian [19] is a Marchaud-type inequality which says that to characterize the k-th modulus of smoothness of a function in Lp (Rs ) in the Lp norm sense, the information of the k-th modulus of smoothness in s independent directions is enough. More precisely, for any vector y in Rs , we denote !k (f; h; y)p := supjtjh krkty f kp ; h > 0. Let yi ; i = 1;


; s be s linearly independent vectors in Rs . Then for any  > 0 and an integer k >  , !k (f; h)p = O(h ) if and only if !k (f; h; yi )p = O(h ) for all i = 1;


; s. Therefore, in Theorem 3.1, the vectors ei ; i = 1;


; s can be replaced by vectors yi ; i = 1;


; s provided that yi ; i = 1;


; s are linearly independent vectors in Rs . For more detail of the above result, the reader is referred to the work of Boman [2] and Ditzian [19, 20]. Based on the above result, the following theorem gives us a characterization of the critical exponent p () of a re nable function  in Lp (Rs ) in terms of its mask provided that the shifts of the re nable function  are stable. Theorem 3.2. Let a function  in Lp(Rs ) (1  p  1) be the normalized solution of the re nement equation (1:1) with a nitely supported re nement mask a P on Zs such that 2Zs a( ) = 2s . For any nonnegative integer k, let k n 1=n k;p (a) := nlim !1 maxf kri Sa kp : i = 1;


; s g:

Then

(3:6)

minf k; p () g  s=p ? log2 k;p (a):

In addition, if the shifts of  are stable, then

(3:7)

minf k; p() g = s=p ? log2 k;p (a):

More generally, let Y := f yi 2 Zs : i = 1;


; s g be a set of s linearly independent vectors. De ne k n 1=n k;p;Y (a) := nlim !1 maxf kryi Sa kp : i = 1;


; s g:

Then the above results still hold true if k;p (a) is replaced with k;p;Y (a).

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Proof. By the de nition of k;p (a), for any real number r such that r > k;p (a), there exists a positive constant Cr such that (3:8) maxf krki San kp : i = 1;


; s g  Cr rn 8 n 2 N : By induction and the de nition of the subdivision operator de ned in (2.2), we observe that X (3:9) rk2?n ei  = rki San ( )(2n
? ); i = 1;


; s: 2Zs Lp (Rs )

Since the function  in is compactly supported, from Equation (3.9), there exists a positive constant C1 depending only on  such that krk2?nei kp  C1 2?ns=p krki San kp 8 n 2 N ; i = 1;


; s: Therefore, it follows from inequality (3.8) that (3:10) krk2?n ei kp  C1 Cr 2?ns=p rn 8 n 2 N ; i = 1;


; s: P On the other hand, by induction, we observe k;p (a)  2s=p?k since 2Zs a( ) = 2s . Therefore, the inequality k  s=p ? log2 k;p (a) holds true for any nonnegative integer k. Since r > k;p (a), we deduce that k  s=p ? log2 k;p?(a) > s=p ? log2 r.
By Theorem 3.1, it follows from inequality (3.10) that  2 Lip s=p ? log2 r; Lp (Rs ) for any r such that r > k;p (a). So in conclusion, we have minf k; p() g  s=p ? log2 k;p (a): If the shifts of the function  are stable, to prove Equation (3.7), it suces to prove that minfk; p ()g  s=p ? log2 k;p (a), equivalently, it suces to prove that k;p (a)  2s=p? for all 0 <  < minf k; p () g: Since the shifts of the function  are stable and  lies in Lp(Rs ), from (3.9), there exists a positive constant C2 depending only on the function  such that krki San kp  C2 2ns=p krk2?nei kp 8 n 2 N ; i = 1;


; s: ?
Since  2 Lip ; Lp(Rs ) and k >  , by Theorem 3.1, we have max f krki San kp g  C2 2ns=p 1max f krk2?nei kp g  C2 C 2n(s=p? ) 8 n 2 N : 1is is

Therefore, the inequality k;p (a)  2s=p? holds true, as desired. The last assertion of this theorem follows directly from Remark 1. Remark 2. If the shifts of the function  are stable and its mask a satis es the sum rules of order k but not k + 1, then p ()  k (see [4, 30]) and therefore, by Theorem 3.2, p () = s=p ? log2 k;p (a): Another remark about the above theorem is that by carefully choosing the set Y , the equality in (3.6) may hold even when the shifts of the function are not stable. For example, let (x) = maxf1 ? jxj=2; 0g; x 2 R: Then the function  is a re nable function with its mask a given by its symbol ea(z ) := 1+(z ?2 +z 2)=2. It is a known fact that the shifts of  are not stable and p () = 1+1=p for any p such that 1  p  1. On the other hand, choose y = 2. It is not dicult to verify that 2;p;y (a) := limn!1 kr2y San k1p=n = 1=2. Therefore, we still have p () = 1=p ? log2 2;p;y (a) = 1=p + 1 for any p such that 1  p  1. In

11

optimal multivariate biorthogonal wavelets

passing, we mention that k;2 (a) can be obtained by nding the spectral radius of a nite matrix by [25, Theorem 4.1]. If k;p (a) < 2s=p for some positive integer k, then 1;p (a) < 2s=p and therefore, by [25, Theorem 3.2] the subdivision scheme associated with the mask a converges in the Lp norm and a 2 Lp(Rs ). Finally, in this section, we prove the following result which will be needed later. Theorem 3.3. Suppose that a function  is a fundamental real-valued function on the real line and  satis es the re nement equation (1:1) with an interpolatory re nement mask a supported on [?3; 3]. Then the inequality 1 ()  2 holds true and therefore,  62 C 2 (R). Proof. We use proof by contradiction to verify our claim. Suppose 1 () > 2. Then a must satisfy the sum rules of order at least 3 (see [4, 30]). By a simple calculation, it is not dicult to see that the symbol ea(z ) can be written as ea(z) = z?3 (1 + z)3 ec(z) with ec(z) := t ? 3 t z + (3=8 + 3t) z2 ? (1=8 + t) z3; for some t 2 R. By [28, Theorem 3.2] or [26, Theorem 3.1], we observe that

?




1=n;

3;1 (a) = 0;1 (c) = nlim !1 max kB1


Bn k : B1 ;


; Bn 2 fA0 ; A1 g where A0 and A1 are matrices given by 0 t 3=8 + 3 t 1 0 A0 := @ 0 ?3 t ?1=8 ? t A ; 0 t 3=8 + 3 t and

0 ?3 t ?1=8 ? t A1 := @ t 3=8 + 3 t 0

?3 t

1

0 0 A: ?1=8 ? t

Therefore, it is evident that n 1=n 3;1 (a) = 0;1 (c)  nlim !1 kA0 k =: (A0 );

where (A0 )pis the spectral radius of A0 . By a simple calculation again, we see that  = 3=16 + (3=8)2 + 4(t + 8t2)=2 is an eigenvalue of A0 . Note that p  = 3=16 + 1=256 + 8(t + 1=16)2  1=4 8 t 2 R: This yields

3;1 (a) = 0;1 (c)  (A0 )  1=4: Since the function  is a fundamental function, the shifts of  are stable. By Theorem 3.2, we have

minf 3; 1 () g = ? log2 3;1 (a)  ? log2 (1=4) = 2: This is a contradiction to our assumption 1 () > 2. Hence, the inequality 1 ()  2 holds true. We are done. 4. Optimal Orthogonal Wavelets in the Multivariate Setting. In [15], Daubechies rst constructed a family of compactly supported orthogonal scaling functions on the real line, namely, Dr (r 2 N ) where Dr satis es the re nement equation (1.1) with the mask Dr supported on [0; 2r ? 1]. It is observed (see [40]) that Dr satfr (z)Dfr (z) = 2ebr (z) for any z in T where br is the is es the sum rules of order r and D

12

bin han

unique univariate interpolatory mask which is supported on [1 ? 2r; 2r ? 1] and satis es the sum rules of order 2r. Therefore, by Corollary 2.4, the mask Dr attains the sum rules of optimal order r. In the multivariate setting, due to the lack of the Riesz factorization theorem, it is much more dicult to construct multivariate orthogonal scaling functions than to construct univariate ones. In the current literature, there are few examples of non-tensor product multivariate orthogonal scaling functions. Before proceeding further, we need the following two lemmas. Lemma 4.1. Let a sequence a on Zs be an interpolatory mask supported on [1 ? 2r; 2r ? 1]s for some positive integer r. De ne a new sequence a1 on Z as follows:

a1 (k) = 21?s

(4:1)

X






2 2Z

X

a(k; 2 ;


; s );

k 2 Z:

s 2Z

If the mask a satis es the sum rules of order at least 2r ? 1, then a1 is a univariate interpolatory re nement mask satisfying the sum rules of order 2r ? 1. Moreover, if the mask a satis es the sum rules of order 2r, then the mask a1 must be the mask br , the unique interpolatory re nement mask which is supported on [1 ? 2r; 2r ? 1] and satis es the sum rules of order 2r. Proof. By the de nition of sum rules given in (1.4), it is easily seen that the sequence a1 satis es the same order of sum rules as the sequence a does. Hence, to complete the proof, it suces to prove that a1 is a univariate interpolatory re nement mask. Namely, we have to prove that a1 (2k) = 0 for all k 2 Znf0g. To this end, it suces to prove that for any " in such that " = (0; "2 ;


; "s ),

X

(4:2)






2 2Z

X

a(2k; 22 + "2 ;


; 2s + "s ) = 0

s 2Z

8 k 2 Znf0g:

Let b be a sequence on Z given by X X b(k) :=


a(2k; 22 + "2 ;


; 2s + "s ); 2 2Z

s 2Z

k 2 Z:

It is evident that b is supported on [1 ? r; r ? 1] since a is supported on [1 ? 2r; 2r ? 1]s. Note that the mask a is an interpolatory re nement mask which satis es the sum rules of order 2r ? 1. By the de nition of sum rules given in (1.4), for any integer j such that 0  j < 2r ? 1, we deduce that X XX X b(k)(2k)j =


a(2k; 22 + "2 ;


; 2s + "s )(2k)j = (j ): k2Z

This gives us (4:3)

k2Z2 2Z r?1 X k=1?r

s 2Z

b(k)kj = (j );

0  j < 2r ? 1:

This linear system has 2r ? 1 unknowns b(1 ? r); : : : ; b(r ? 1) and 2r ? 1 equations, and its coecient matrix is a Vandermonde matrix. Hence, it has a unique solution. It is easily seen that b(j ) = (j ), j = 1 ? r; : : : ; r ? 1 is a solution to the above linear system. This veri es (4.2), thereby completing the proof. Lemma 4.2. Let a function  in Lp (Rs ) (1  p  1) be the normalized solution of the re nement equation (1:1) with a nitely supported re nement mask a on Zs. Let the sequence a1 be given by Equation (4:1) and a1 be the normalized solution of the re nement equation (1:1) with the re nement mask a1 . Suppose that the shifts of

13

optimal multivariate biorthogonal wavelets

 are stable. Then the subdivision scheme associated with the mask a1 converges in the Lp norm and p ()  p (a1 ). Proof. In the following, we shall prove that for any nonnegative integer k, there exists a positive constant C such that (4:4) krk1 San1 kp  C 2n(1?s)=p krk1 San kp 8 n 2 N :

n From the de nition of the subdivision operator given in (2.2), we observe that Sg a  (z ) = Q n?1 e 2 s n n g 2(1?s)nSg a  (z; 1;


; 1) j =0 a(z ) for any z in T . Therefore, we deduce Sa1  (z ) = Q n ? 1 2 1 ? s n for any z in T since ae1 (z ) = 2 ea(z; 1;


; 1) and Sg a1  (z ) = j =0 ae1 (z ) for any z in T. That is, X n (1?s)n X n j

j






Sa1 (j ) = 2

(4:5)

2 2Z

Sa (j; 2 ;


; s )

s 2Z

8 j 2 Z; n 2 N :

Since r1 ( ) = ( ) ? ( ? e1);  2 `0 (Zs) where e1 is the rst coordinate unit vector, we have

rk1 San1 (j ) = 2(1?s)n rk1 = 2(1?s)n

X






X

San(j; 2 ;


; s )

X 22Z X 2Zk




2 2Z

s

r1 San(j; 2 ;


; s ):

s 2Z

Since the mask a is nitely supported, there exists a positive integer r such that supp a  [?r; r]s . It is easily seen that suppSan   [?2nr; 2n r]. Therefore, the above equality can be rewritten as

rk1 San1 (j ) = 2(1?s)n

2 r X n

2 =?2

nr






2 r X n

s =?2

nr

rk1 San (j; 2 ;


; s );

j 2 Z:

Applying the Holder inequality to the above sum, we obtain

jrk1 San1 (j )jp  2n(1?s)p (2n+1 r + 1)(s?1)p=q  C1

2n(1?s)

X






2 2Z

X

X






X

jrk1 San (j; 2 ;


; s )jp

2 2Z s 2Z k jr1 San (j; 2 ;


; s )jp ;

s 2Z

where 1=p + 1=q = 1 and C1 = (2r + 1)(s?1)p=q . It follows from the above inequality that krk1 San1 kp  C11=p 2n(1?s)=p krk1 San kp 8 n 2 N : Therefore, the inequality (4.4) holds true. Since the shifts of  are stable and  lies in Lp (Rs ), the subdivision scheme associated with the mask a converges in the Lp norm. That is, by [25, Theorem 3.2], it is equivalent to lim maxf kri San k1p=n : i = 1;


; s g < 2s=p :

n!1

The reader is referred to [25] for detailed discussion on the convergence of a subdivision scheme in the Lp norm. Taking k = 1 in (4.4), we get n 1=n 1=p lim kr1 San1 k1p=n  2(1?s)=p nlim !1 maxf kri Sa kp : i = 1;


; s g < 2 :

n!1

14

bin han

Hence, the subdivision scheme associated with the mask a1 converges in the Lp norm. In particular, we have a1 2 Lp (R). Note that k;p (a1 ) := limn!1 krk1 San1 k1p=n and k n 1=n k n 1=n k;p (a) := nlim !1 maxf kri Sa kp : i = 1;


; s g  nlim !1 kr1 Sa kp :

Hence, the inequality (4.4) gives rise to

k;p (a1 )  2(1?s)=p k;p (a) 8 k 2 N [ f0g: Let k be a positive integer greater than p (). It follows from Theorem 3.2 that p (a1 )  1=p ? log2 k;p (a1 )  s=p ? log2 k;p (a) = p (); as desired. Combining the above Lemmas and Theorem 3.3, we have the following result: Corollary 4.3. Suppose that a function  is a fundamental real-valued function and satis es the re nement equation (1:1) with an interpolatory re nement mask a supported on [?3; 3]s. Then the inequality 1 ()  2 holds true and therefore,  does not belong to C 2 (Rs ). Proof. Let the sequence a1 on Z be given in (4.1). Suppose that 1 () > 2. Then the mask a must satisfy the sum rules of order at least 3. Therefore, it follows from Lemma 4.1 that a1 is an interpolatory mask. Let a1 be the normalized solution of (1.1) with the mask a1 . Then by Lemma 4.2, the subdivision scheme associated with a1 converges in the L1 norm which implies that the function a1 is a fundamental function. From Lemma 4.2, we also have 1 ()  1 (a1 ). It follows from Theorem 3.3 that 1 ()  1 (a1 )  2. This is a contradiction to our assumption 1 () > 2. Therefore, the inequality 1 ()  2 holds true. The above Corollary 4.3 says that there is no C 2 fundamental re nable function supported on [?3; 3]s. This result also implies that if a function  is an orthogonal scaling function supported on [0; 3]s, then 2 ()  1 and therefore,  62 C 1 (Rs ). Let  be an orthogonal scaling function with its mask supported on [0; 2r ? 1]s for some positive integer r. From Corollary 2.4, we see that S () can provide approximation order at most r. For this case, we shall study the upper bound of the critical exponent p () for any p such that 1  p  1. Based on the above lemmas and Theorem 3.2, we have the following result on orthogonal scaling functions. Theorem 4.4. Suppose that a function  in L2 (Rs ) is an orthogonal scaling function with its re nement mask a supported on [0; 2r ? 1]s \ Zs for some positive integer r. De ne a new sequence a1 on Z as follows:

a1 (k) := 21?s

X






2 2Z

X

a(k; 2 ;


; s );

s 2Z

k 2 Z:

Let a1 be the normalized solution of the re nement equation (1:1) with the mask a1 . If the mask a satis es the sum rules of optimal order r, then the function a1 is an orthogonal scaling function with the mask a1 satisfying

ae1 (z ) ae1 (z ) = 2ebr (z ); z 2 T: If in addition, the function  belongs to Lp (Rs ) for some p such that 1  p  1, then p ()  p (a1 ):

15

optimal multivariate biorthogonal wavelets

In particular,

2 ()  2 (Dr ) and 2 (a1 ) = 2 (Dr ) = 1 (br )=2;

where Dr is the Daubechies orthogonal scaling function with its mask Dr supported on [0; 2r ? 1], and br is the Deslauriers and Dubuc fundamental re nable function with its mask br supported on [1 ? 2r; 2r ? 1]. Proof. Let a sequence b on Zs be given by its symbol eb(z) := 2?sea(z)ea(z); z 2 Ts: By Lemma 2.1, the sequence b is an interpolatory re nement mask since  is an orthogonal scaling function. Since the mask a satis es the sum rules of order r, by the proof of Theorem 2.3, we see that the sequence b must satisfy the sum rules of order at least 2r. De ne a new sequence c on Z as in Equation (4.1) by

c(k) = 21?s

X






2 2Z

X

b(k; 2 ;


; s );

s 2Z

k 2 Z:

By Lemma 4.1, the sequence c must be the mask br since the sequence b is supported on [1 ? 2r; 2r ? 1]s and satis es the sum rules of order 2r. Note that ec(z ) = 21?seb(z; 1;


; 1) and ae1 (z ) = 21?sea(z; 1;


; 1) for any z 2 T. Therefore,

ae1 (z )ae1 (z ) = 22?seb(z; 1; : : :; 1) = 2ec(z ) = 2ebr (z ) 8 z 2 T: Thus, the mask a1 is the dual mask of itself for s = 1. By Lemma 4.2, the subdivision scheme associated with the mask a1 converges in the L2 norm since the function  is a scaling function. Hence, the function a1 is an orthogonal scaling function by Lemma 2.1. If  lies in Lp(Rs ) for some p such that 1  p  1, then by Lemma 4.2, we have p ()  p (a1 ). Note that ae1 (z )ae1 (z ) = 2ebr (z ) implies that 2 (a1 ) = 1 (br )=2. fr (z)Dfr (z) = 2br (z) for any z in T, Since D 2 ()  2 (a1 ) = 1 (br )=2 = 2 (Dr ) which completes the proof. In the following, we give an example to demonstrate that when s > 1, such optimal orthogonal scaling functions are not unique. Example 4.5. The mask a is supported on [0; 3]2 and is given by

0 3 p?10+6 p3 p3 BB ? 8 ? p 8 p + p8 BB ? 81 + 83 + ?10+6 3 BB p p 8 p BB 58 + 83 + ?10+6 3 @ p p 8 p

1 4 1 2 1 4

p

p

p ?10+6 3 5 3 3 4 8? 8 + 8 p?10+6 p3 p 1 7?3 3? 2 8 8 8 p3 1 p?10+6 p3 p3 + 2 8? + 8 8 p p3 p p + 43 ? 81 + 83 + ?10+6 8

?

p3

1 C p C 1? 3C 4 4 C C p C: 1? 3C 4 4 C A 0

3 + 83 ? ?10+6 0 8 Then the function a is an orthogonal scaling function and the mask a satis es the sum rules of order 2. Moreover, by calculation, we have 2 (a ) = 1. Combining Theorem 4.4 and Corollary 4.3, we see that for any orthogonal scaling  with its mask supported on [0; 3]s , the inequality 2 ()  1 holds true. Therefore, the function a is an optimal orthogonal scaling function in the L2 norm sense. The graph and contour of a are presented in Figure 1. 3 8

16

bin han

3

1.5

2.5

1 2

0.5 1.5

0 1

−0.5 3 2.5

3

2

2.5 1.5

0.5

2 1.5

1

1

0.5

0.5 0

0

0

0

0.5

1

1.5

2

2.5

Figure 1. The graph and contour of the orthogonal scaling function a in Example 4.5.

5. Optimal Multivariate Biorthogonal Wavelets. In this section, we will demonstrate a result similar to Theorem 4.4 for the biorthogonal wavelets. Since there is a close relation between a biorthogonal wavelet and a fundamental re nable function, let us rst prove the following result on fundamental re nable functions: Theorem 5.1. Let  be a fundamental re nable function with a nitely supported interpolatory mask a. Suppose that a is supported on [1 ? 2r; 2r ? 1]s for some positive integer r and the mask a satis es the sum rules of order 2r ? 1. Let a sequence a1 on Z be given by Equation (4:1) and let a1 be the normalized solution of the re nement equation (1:1) with the mask a1 . Then the function a1 is a fundamental function and p ()  p (a1 ) 8 1  p  1: Moreover, if the mask a satis es the sum rules of order 2r, then p ()  p (br ) 8 1  p  1: In other words, the inequality p ()  p (tr ) holds true where tr is the tensor product interpolatory mask given in (2:7). Proof. By Lemma 4.1, we see that the mask a1 is an interpolatory re nement mask. Since the function  is fundamental, the shifts of  are stable. By Lemma 4.2, the subdivision scheme associated with the mask a1 converges in the Lp norm for any p such that 1  p  1. Hence a1 , the normalized solution of the re nement equation (1.1) with the interpolatory re nement mask a1 , is continuous and therefore fundamental. It follows from Lemma 4.2 that p ()  p (a1 ) for any 1  p  1. If the mask a satis es the sum rules of order 2r, by Lemma 4.1, then the sequence a1 must be the mask br . Hence, by Lemma 4.2, p ()  p (br ) for any p such that 1  p  1. The reader is referred to [18, 21, 22, 26, 40, 41] on interpolatory subdivision schemes. In particular, a general construction of bivariate interpolatory masks gr (r 2 N ) was reported by Han and Jia in [26] with each mask gr supported on [1 ? 2r; 2r ? 1]2 , satisfying the optimal sum rules of order 2r and 2 (gr ) = 2 (br ) at least for r = 1;


; 12. Recall that by a we denote the normalized solution of the

3

optimal multivariate biorthogonal wavelets

17

re nement equation (1.1) with a mask a. A similar result to Theorem 4.4 for a biorthogonal wavelet is the following: Theorem 5.2. Let a function  in L2 (Rs ) be a scaling function with a re nement mask a, and a function d in L2 (Rs ) be a dual scaling function of  with a re nement mask ad . De ne two new sequences a1 and ad1 on Z as follows: X X a1 (k) = 21?s


a(k; 2 ;


; s ); k 2 Z; and

2 2Z

ad1 (k) = 21?s

X






2 2Z

s 2Z

X

ad(k; 2 ;


; s );

s 2Z

k 2 Z:

By a1 and ad1 we denote the normalized solutions of the re nement equation (1:1) with the masks a1 and ad1 respectively. Let a sequence b on Zs be given as in (2:4) by X a( ? )ad( );  2 Zs: (5:1) b() := 2?s 2Zs

Suppose that the sequence b is supported on [1 ? 2k; 2k ? 1]s \ Zs for some positive integer k and b satis es the sum rules of order 2k ? 1. Then the function a1 is a univariate scaling function with ad1 being a dual scaling function of a1 . If  belongs to Lp (Rs ) and d belongs to Lq (Rs ) for some p; q such that 1  p; q  1, then a1 2 Lp(R), ad1 2 Lq (R) and

(5:2) p ()  p (a1 ) and q (d )  q (ad1 ): In particular, if the sequence b satis es the sum rules of order 2k, then ae1 (z )aed1 (z ) = 2bek (z ); z 2 T and q (d )  r (bk ) ? p (); where 1=r = 1=p + 1=q ? 1 and bk is the unique interpolatory mask which is supported on [1 ? 2k; 2k ? 1] and satis es the sum rules of order 2k. Proof. By Lemma 2.1, it is easily seen that the sequence b is an interpolatory mask. Let c be a sequence on Z given by X X c(k) = 21?s


b(k; 2 ;


; s ); k 2 Z: 2 2Z

s 2Z

It follows from Lemma 4.1 that the sequence c is an interpolatory mask since the sequence b is supported on [1 ? 2k; 2k ? 1]s and satis es the sum rules of order 2k ? 1. Observe that ec(z ) = 21?seb(z; 1


; 1), ae1 (z ) = 21?sea(z; 1;


; 1) and aed1 (z ) = 21?s aed (z; 1;


; 1) for any z in T. It is easy to see that

(5:3) ae1 (z )aed1 (z ) = 22?2sea(z; 1


; 1)aed(z; 1;


; 1) = 2ec(z ); z 2 T: Therefore, the masks a1 and ad1 must satisfy the discrete biorthogonal relation (2.5) with s = 1 since the sequence c is an interpolatory mask. Since both  and d belong to L2 (Rs ) and their shifts are stable, by Lemma 4.2, the subdivision schemes associated with the masks a1 and ad1 converge in the L2 norm respectively. Thus, by Lemma 2.1, the function a1 is a scaling function with ad1 being a dual scaling function of a1 . The inequality (5.2) follows directly from Lemma 4.2. If the sequence b satis es the sum rules of order 2k, by Lemma 4.1, the mask c in k1 S n  (z ) = (1 ? z )k1 n?1 ae1 (z 2j ). Equation (5.3) must be the mask bk . Note that r^ a1 j =0

18

bin han

Therefore, it follows from (5.3) that for any positive integers k1 and k2 , it is easy to verify that k2 S nd  (z ); k1 S n  (z )r^ +k2 S n  (z ) = r^ z 2 T: 2n rk1^ a1 bk a1

Therefore, by applying Young's inequality to the above equation, we have 2n krk1 +k2 Sbnk kr  krk1 San1 kp krk2 Sand1 kq 8 n 2 N ;

where 1=r = 1=p + 1=q ? 1. This yields 2k1 +k2 ;r (bk )  k1 ;p (a1 )k2 ;q (ad1 ) 8 k1 ; k2 2 N : By Theorem 3.2, we have r (bk )  p (a1 ) + q (ad1 ). Since p ()  p (a1 ) and q (d )  q (ad1 ), we conclude that r (bk )  p (a ) + q (ad ). Corollary 5.3. Let  be a scaling function with a re nement mask a supported on [?l; l]s for some positive integer l, and d be a dual scaling function with a re nement mask ad supported on [1 + l ? 2k; 2k ? l ? 1]s for some positive integer k. Let the sequence b be given in (5:1). Suppose that the mask a satis es the sum rules of order m. Then the mask ad can satisfy the sum rules of order at most 2k ? m. Moreover, if the mask ad satis es the sum rules of order 2k ? m ? 1 (or 2k ? m), then the sequence b can satisfy the sum rules of order at least 2k ? 1 (or 2k) and the corresponding results in Theorem 5:2 hold true. Proof. This is a direct consequence of Theorem 2.3 and Theorem 5.2. Let us consider an example. Let  be a re nable box spline function with its mask a given by its symbol ea(z) = 2?ssj=1 (zj?1 + 2 + zj ); z 2 Ts or ea(z) = 2?1(1 + z1?1


zs?1)sj=1 (1 + zj ); z 2 Ts: It is easy to verify that  is a fundamental function with 1 () = 2, its mask a is supported on [?1; 1]s and a satis es the sum rules of order 2. Thus, the function  is a scaling function. Then Corollary 4.3 and Corollary 5.3 imply that if a function d is a dual scaling function of the scaling function  with its mask supported on [?2; 2]s, then the function d can not be continuous. For any dual scaling function d of the scaling function  with its mask ad supported on [2 ? 2r; 2r ? 2]s for some positive integer r, by Theorem 2.3, the mask ad can satisfy the sum rules of order at most 2r ? 2. If ad satis es the sum rules of order 2r ? 2, by Corollary 5.3, then we have 2 (d )  2 (br ) ? 1 () = 2 (br ) ? 2: When s = 2, in the next section, we shall construct a family of dual scaling functions Hr (r 2 N ) of the bivariate hat function  such that the dual mask Hr is supported on [2 ? 2r; 2r ? 2]2 and satis es the sum rules of order 2r ? 2. In addition, the equality 2 (Hr ) = 2 (br ) ? 2 holds true at least for r = 3;


; 12 and each mask Hr is symmetric about the two coordinate axes, and the lines x1 = x2 and x1 = ?x2 . 6. Construction of Multivariate Biorthogonal Wavelets. In this section, we shall present a general method to construct multivariate biorthogonal wavelets. More precisely, for any scaling function  with an interpolatory re nement mask a, a general CBC (Construction By Cosets) algorithm is given to produce all the dual masks of the mask a. As an application of this general theory, for any bivariate

optimal multivariate biorthogonal wavelets

19

fundamental mask a which is symmetric about the two coordinate axes, we construct a family of dual masks of a which satisfy any desired order of sum rules and are also symmetric about the two coordinate axes. Based on this construction, a family of optimal bivariate biorthogonal wavelets is presented. Such biorthogonal wavelets have full symmetry (i.e., they are symmetric about the x1 -axis, x2 -axis, and the lines x1 = x2 and x1 = ?x2 ), have the optimal order of sum rules, the optimal L2 smoothness order and relatively small support of the dual masks. Before proceeding further, we introduce some notation. Recall that

f (1 ;


; s ) 2 Zs : i  0 8 i = 1;


; s g: For any  = (1 ;


; s ) 2 Zs, we denote jj := j1 j +


+ js j and ! := 1 !


s ! if  2 Zs+. For any  = (1 ;


; s );  = (1 ;


; s ) 2 Zs, by    we mean i  i for all i = 1;


; s, and by  <  we mean    and  = 6 . Throughout this section, for any  2 Zs+, by p we denote the monomial (
) and Zs+ :=

h; p i :=

X

2Zs

()p () =

X

2Zs

() ;

Theorem 6.1. Let a sequence a on Zs satisfy

sequence relation

ad

on

Zs

P

2Zs a( ) = 2

s.

Suppose that a is a dual mask of a that satis es the following discrete biorthogonal

X

(6:1)

 2 `0 (Zs):

2Z

s

a( ? 2)ad ( ) = 2s ()

8  2 Zs:

If the sequence ad satis es the sum rules of order k for some positive integer k, then for any  2 Zs+ such that jj < k, the value h := 2?shad ; p i is uniquely determined by the sequence a. More precisely, h is given by the following recursive relation: X j?j ! ha; p? i h ; jj < k;  2 Zs+: (6:2) h = () ? 2?s (?1)  !(  ?  )! 0 0 g: Now we set ad (2 + (1; 0)) = 0 for any 2 Z2+n(E [ F ) and ?
ad (2 1 + 1; 2 2) = 1 + ( 2 ) c( 1 ; 2 ) ; ( 1 ; 2 ) 2 E [ F with some yet-to-be-determined parameters c ; 2 E [ F . This extra freedom c ; 2 F given by F will be used to reduce the support of the mask ad at the coset (0; 0) constructed in Step (7) of the TCBC algorithm. More precisely, we try to adjust the coecients of Hk?1 on the set f( 1 ; 2 ) 2 Z2 : 1 + 2 = 2k ? 2g to be zero. By using symmetry, after a simple calculation, it is easily seen that this restriction is equivalent to the following linear system c( 1 ; 2) + c( 2 ?1; 1 +1) + b( 1 ; 2 ?1) =2 = 0 for all ( 1 ; 2 ) 2 F: By simply setting c( 1 ; 2 ) = 0 for any ( 1 ; 2 ) 2 F such that k=2  1 < k, the above linear system has a unique solution c ; 2 F . Now the following linear system

X ? X ?

c 2 + (1; 0) 2 = ha (2)=4 ? c 2 + (1; 0) 2 ;

2E

2F

jj < k;  2 Z2+

has a unique solution for c ; 2 E by Lemma 6.2. Set ad (2 1 ; 2 2 + 1) = ad (2 2 + 1; 2 1); ( 1 ; 2 ) 2 Z2+. By the TCBC algorithm, we have a dual mask ad of the mask ah such that ad satis es the sum rules of order 2k. We shall use Hr to denote the dual mask of the mask ah derived from the above modi ed TCBC algorithm such that Hr satis es the sum rules of order 2r ? 2. For each positive integer r, by Gr we denote the following set Gr := f(1 ; 2 ) 2 Z2 : j1 j + j2 j = 2r ? 1 and either j1 j or j2 j is an even number less than r ? 1g: To sum up and restate the above construction of the dual masks Hr of the mask ah , we have the following theorem. Theorem 6.4. Let r be a positive integer greater than two. Then there exists a unique re nement mask Hr satisfying the following conditions: (1) Hr is supported on  ( ;  ) 2 Z2 : j j + j j  2r ? 1; maxf j j; j jg  2r ? 2 nG ; 1 2 1 2 1 2 r (2) Hr is symmetric about the two coordinate axes, the lines x1 = x2 ; x1 = ?x2 ; (3) Hr satis es the sum rules of order 2r ? 2; (4) Hr and ah (the mask ah is given in (6:11)) satisfy the dual relation (6:1). Remark 3. The set Gr appears strange. The reason is that in our modi ed TCBC algorithm, we set c( 1 ; 2 ) = 0 for any ( 1 ; 2 ) 2 F such that (r ? 1)=2  1 < r ? 1. Note that both Hr and Hr are symmetric about the x1 -axis, x2 -axis, and the lines x1 = x2 and x1 = ?x2 . Let a be a multivariate interpolatory mask such that a satis es the sum rules of order k. For any positive integer r, ?by convolution, it is
easy to obtain a new interpolatory mask b such that eb(z ) = ea(z ) r cer (z ); z 2 Ts where cer (z ) can be explicitly expressed by using ea(z ). Such interpolatory mask b satis es the sum rules of order rk by Theorem 2.2. See Proposition 3.7 in Han [24]

25

optimal multivariate biorthogonal wavelets

for detailed discussion on construction of biorthogonal wavelets using this convolution method. Such method was further discussed by Ji, Riemenschneider, and Shen [27]. The TCBC algorithm proposed in this paper can be generalized to the general case and it has many advantages over the convolution method. We shall illustrate the advantages of our CBC and TCBC algorithms over the convolution method and other known methods in the literature on construction of biorthogonal wavelets elsewhere. Let us provide detail in the following for the masks H3 and H4 . Example 6.5. The mask H3 is supported on [?4; 4]2 and is given by 00 0 0 3 3 3 0 0 01 C BB 0 0 0 ?1283 643 1283 0 0 C C BB 64 ? 32 ? 64 0 BB 0 0 161 ? 18 ? 83 ? 81 161 0 0 CCC

BB 3 3 1 11 51 11 1 3 BB 128 ? 64 ? 8 32 64 32 ? 8 ? 64 BB 3 ? 3 ? 3 51 33 51 ? 3 ? 3 BB 643 323 81 1164 1651 6411 81 323 BB 128 ? 64 ? 8 32 64 32 ? 8 ? 64 BB 0 0 1 ? 1 ? 3 ? 1 1 0 16 8 8 8 16 BB 3 3 B@ 0 0 0 ? 64 ? 32 ? 643 0 0

C

3 C 128 C C 3 C : 64 C C C 3 128 C C

C

0 C CC 0 C A 3 3 3 0 0 0 128 64 128 0 0 0 Then the mask H3 satis es the sum rules of order 4 and H3 is a dual scaling function of 'h with 2 (H3 )  1:17513. Thus, the function H3 is an optimal dual scaling function of the function 'h in the L2 norm sense since 2 (H3 )  2 (b3 ) ? 1 ('h ). The graphs and contours of the scaling function 'h and the dual scaling function H3 with their associated wavelets are given at the end of this paper. Example 6.6. The mask H4 is supported on [?6; 6]2 and the part of H4 in the rst quadrant is supported on [0; 6]2 and is given by 0? 5 ? 5 0 0 0 0 0 1 512 1024 BB 5 5 CC 0 0 0 0 0 C BB 256 512 83 15 9 145 BB 1024 4096 ? 2048 ? 4096 0 0 0 CCC BB ? 363 ? 87 15 C 9 9 ? 4096 0 0 C BB 2048 1024 1024 1024 CC 359 ? 69 1 15 15 BB ? 1024 0 C C 512 16 1024 ? 2048 0

BB @

C

69 ? 87 145 5 5 C ? 512 1024 4096 512 ? 1024 A 359 ? 363 5 83 5 ? 1024 2048 1024 256 ? 512 with the at the bottom-left as H4 (0; 0). Since H4 is symmetric about the coordinate axes, other parts of H4 are obtained by symmetry as in (6.10). By calculation, we have 2 (H4 )  1:79313 and the mask H4 satis es the sum rules of 1723 2048 493 256 493 number 256

401 1024 1723 2048

order 6. Thus, the function H4 is an optimal dual scaling function of 'h in the L2 norm sense since 2 (H4 )  2 (b4 ) ? 1 ('h ). The graph and contour of H4 are presented in Figure 2. Recall that by br we denote the interpolatory re nement mask supported on [1 ? 2r; 2r ? 1] as constructed by Deslauriers and Dubuc in [18]. By adtr we denote

26

bin han

6

3 4

2.5 2 1.5

2

1 0.5

0

0 −0.5 −2

−1 6 4

6

2

4 0

−4

2 0

−2

−2

−4

−4 −6

−6 −6

−6

−4

−2

0

2

4

Figure 2. The graph and contour of the function H4 .

the tensor product dual scaling function of 'h with its mask adtr satisfying

afh(z1 ; z2)afdtr (z1 ; z2 ) = ber (z1 )ber (z2 ); (z1 ; z2 ) 2 T2 : Let the masks Hr and Hr be the dual masks constructed by the TCBC algorithm and the modi ed TCBC algorithm respectively such that both Hr and Hr satisfy the sum rules of order 2r ? 2. In the following, we use N (a) to denote the number of nonzero coecients in the re nement mask a. The values of 2 (br ) are taken from [23]. The following table shows that for r = 3;


; 12, the function Hr is an optimal dual scaling function of 'h in the L2 norm sense. r

3 4 5 6 7 8 9 10 11 12

2 (br ) 3:17513 3:79313 4:34408 4:86202 5:36283 5:85293 6:33524 6:81144 7:28260 7:74953

2 (adtr ) 1:17513 1:79313 2:34408 2:86202 3:36283 3:85293 4:33524 4:81144 5:28260 5:74953

2 (Hr ) 0:42927 0:98084 1:46708 1:90387 2:30033 2:66264 2:99578 3:30381 3:58991 3:85672

2 (Hr ) N (adtr ) N (Hr ) N (Hr ) 1:17513 81 49 49 1:79313 169 97 101 2:34408 289 161 161 2:86202 441 241 245 3:36283 625 337 337 3:85293 841 449 453 4:33524 1089 577 577 4:81144 1369 721 725 5:28260 1681 881 881 5:74953 2025 1057 1061

Acknowledgment: The author is indebted to Professor Rong-Qing Jia for his advice and comments on this paper, and to Professor Zeev Ditzian for providing his reprints to and sharing his ideas with the author.

6

27

optimal multivariate biorthogonal wavelets

1 3 2.5

0.8

2 0.6

1.5 1

0.4

0.5 0.2

0 −0.5

0 1

3 0.5

2

1

3

1

0.5

0

2

0

1

0 −0.5

−1

−0.5 −1

0 −2

−1 −3

−1

−2

3.5

3 2.5

3

2

2.5

1.5

2

1

1.5

0.5

1

0

0.5

−0.5

0

3

3 2

3 2

1 1

0

0 −1

−1

−2 −2

−3

2

3 1

2 1

0 0

−1

−1 −2

−2

Figure 3. (a) is the scaling function 'h and (b), (c) and (d) are the associated three wavelets 1 , 2 and 3 in Example 6.5.

28

bin han

6

8

5

6

4 4 3 2

2

1

0

0

−2

−1

−4

−2 4

2 2

4

1

3

2

0

2

0

1

0 −2

−2 −4

0

−1 −1 −2

−4

8

−2

5

6

4

4

3

2 2 0 1 −2 0 −4 −1 3

3 2

2 1

1 0

0 −1

−1 −2

−2

2

3 1

2 1

0 0

−1

−1 −2

−2

Figure 4. (a) is the dual scaling function H3 and (b), (c) and (d) are the associated three dual wavelets 1d , 2d and 3d in Example 6.5.

29

optimal multivariate biorthogonal wavelets 1

3 2

0.5 1 0

0 −1

−0.5 −2 −1 −1

−0.5

0

0.5

1

−3 −2

3

3

2

2

1

1

0

0

−1

−1

−2 −3

−2

−1

0

1

2

3

−2 −2

4

2

2

1

0

0

−2

−1

−4 −4

−2

0

2

4

−2 −2

3

3

2

2

1

1

0

0

−1

−1

−2 −2

−1

0

1

2

−2 −2

−1

0

1

2

3

−1

0

1

2

3

−1

0

1

2

3

−1

0

1

2

3

Figure 5. The contours of the scaling function 'h ,its dual scaling function H3 and the associated wavelets and dual wavelets in Example 6.5.

30

bin han REFERENCES

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