Multiwavelet Frames From Refinable Function Vectors ¤ Bin Han and Qun Mo Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1 E-mail: fbhan, mog@math.ualberta.ca Abstract Starting from any two compactly supported d-refinable function vectors in ¡L2(R)¢r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5, 12, 13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper. Key words: Dual wavelet frames, wavelet frames, refinable function vectors, multiwavelets, refinable Hermite interpolants, sum rules, vanishing moments, symmetry. 2000 AMS subject classification: 42C40, 42C15, 41A15, 41A25 . ¤Research supported in part by the Natural Sciences and Engineering Research Council (NSERC) Canada under Grant G121210654 and by Alberta Innovation and Science REE under Grant G227120136. Web: http://www.ualberta.ca/»bhan 1
1 Introduction Wavelet frames and multiwavelets have been extensively studied in the literature, for example, to mention only a few here, see [1]—[32] and numerous references therein. Recently, there is a growing interest in both theory and application to study and construct redundant systems such as wavelet frames due to several attractive features of a…