Wavelets with Short Support

Abstract

This paper is to construct Riesz wavelets with short support. Riesz wavelets with short support are the objective of interest in both theory and application. In theory, it is known that a B-spline of order m has the shortest support among all compactly supported refinable functions with the same regularity. However, it remained open whether a Riesz wavelet with the shortest support and m vanishing moments can be constructed from the multiresolution analysis generated by the B-spline of order m. In various applications, a Riesz wavelet with a short support, a high order of regularity, and vanishing moments is often desirable in signal and image processing, since they have a good time frequency localization and approximation property, as well as fast algorithms. This paper presents a theory for the construction of Riesz wavelets with short support and gives various examples. In particular, from the multiresolution analysis whose underlying refinable function is the B-spline of order m, we are able to construct the shortest supported Riesz wavelet with m vanishing moments. The support of the wavelet functions can be made even shorter by reducing their orders of vanishing moments. The study here also provides a new insight into the structures of the spline tight frame systems constructed in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408-447, I. Daubechies, B. Han, A. Ron, and Z. Shen, Appl. Comput. Harmon. Anal., 14 (2003), pp. 1-46, B. Han and Q. Mo, Proc. Amer. Math. Soc., 132 (2004), pp. 77-86] and bi-frame systems in [I. Daubechies, B. Han, A. Ron, and Z. Shen, Appl. Comput. Harmon. Anal., 14 (2003), pp. 1-46, I. Daubechies and B. Han, Constr. Approx., 20 (2004), pp. 325-352].