Orthogonal Multiwavelets of Multiplicity Four

K e y w o r d s W a v e l e t s , Multiwavelets, Approximation order, Refinement equations, Subdivision. 1. I N T R O D U C T I O N Wavele t t heo ry is based on the idea of mul t i r eso lu t ion analys is (MRA) . Usually, an M R A is gene r a t e d by one scal ing funct ion. However, such wavelets canno t possess t he p rope r t i e s of shor t s u p p o r t , s y m m e t r y or an t i symmet ry , and o r thogona l i t y s imul taneously . T h e s t u d y of mul t i wavele ts was i n i t i a t ed by G o o d m a n , Lee and Tang in [1]. Then G o o d m a n and Lee in [2] gave a c h a r a c t e r i z a t i o n of scal ing funct ions and wavelets. Mul t iwavele t s open new poss ib i l i t ies in t he cons t ruc t i on of wavelets wi th those p rope r t i e s based on mul t i sca l ing funct ions. Mul t iwave le t s have more f reedom in the i r cons t ruc t ion . Therefore , t hey can have shor te r s u p p o r t w i th more van ish ing m o m e n t s t h a n a single wavelet , and t hey m a y have o r t hogona l i t y and s y m m e t r y a t t he s ame t ime. These p rope r t i e s are very des i rable in m a n y appl ica t ions . Thus , mul t iwave le t s can be ve ry useful for var ious p rac t i ca l p rob lems (see [3,4], for examples) . T h e l i t e r a tu re on th is sub jec t is growing r ap id ly (see [5] and the references there in) . In th is paper , we consider so lu t ions of a The authors are very grateful to the anonymous referees for their valuable comments and suggestions. They also would like to thank L. Guan and P. Xiao for their valuable discussions during the revision of the paper and for the numerical computation about the joint spectral radius. The research was supported in part by an RDC grant of ETSU under Grant ~00-007/m. 0898-1221/00/$ see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by ,4Az~S-TFjX PII: S0898-1221 (00) 00229-7 1154 D. HONG AND A.-D. Wu system of refinement (scaling) equations in the form 2 ¢(x) = hk¢ (2x k) , k=0 where ¢ is a 4 x 1 vector function, and h0, hi, and h2 are 4 x 4 matrices (the sequence (hk) of matrices is called a refinement mask). We give explicit expressions of hk, k = O, 1, 2 such that the refinement (scaling) function vector ¢ has properties of short support [0, 2], symmetry or antisymmetry and orthogonality. The multiwavelets with the same properties as the scaling functions are also constructed. The paper is organized as follows. In the remainder of this section, we recall some results on multiwavelets construction. In Section 2, we construct a scaling function vector with multiplicity 4 such that it has support on [0, 2], symmetry or antisymmetry, and orthogonality. The multiwavelets with the same properties as the scaling functions are constructed in Section 3. We discuss the properties of approximation order and analyze the smoothness of the refinement functions in the final section. In general, we are concerned with the system of refinement equations N ¢(X) = Z hk¢ (2x -k) , (1.1) k=0 where h k i s an r x r matrix, and ¢ = ( ¢ 1 , ¢ 2 , . . . , ¢ r ) T is a vector of functions. transform of the above refinement equation is