Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix

Authors:
Bin Han
Affiliations:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1
Venue:
Journal of Computational and Applied Mathematics - Special issue: Approximation theory, wavelets, and numerical analysis
Year:
2003

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Abstract

Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight M-wavelet frames and orthonormal M-wavelet bases in L2(Rd) of exponential decay, which are derived from compactly supported M-refinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several results in Battle (Comm. Math. Phys. 110 (1987) 601), Bownik (J. Fourier Anal. Appl. 7(2001) 489), Gröchenig and Ron (Proc. Amer. Math. Soc. 126 (1998) 1101), Lemarié (J. Math. Pures Appl. 67 (1988) 227), and Strichartz (Constr. Approx. 9 (1993) 327).