SMOOTH REFINABLE FUNCTIONS AND WAVELETS OBTAINED BY CONVOLUTION PRODUCTS

This paper is concerned with the construction of smooth refinable func tions relative to a large class of expanding scaling matrices. Charact eristic functions of certain self-similar tiles related to a given sca ling matrix are the simplest examples of such refinable functions. It is known that a sufficiently high convolution power of such a characte ristic function produces eventually refinable functions of arbitrary h igh regularity. The objective of this paper is to quantify the number of convolutions needed to achieve continuous differentiability. This t urns out to be possible when using convolutions of possibly different judiciously chosen tiles associated with the same scaling matrix. An e ssential ingredient of our analysis is the concept of stationary subdi vision schemes which allows us to derive explicit estimates for the sm oothness of the resulting convolution products. Once a regular refinab le function is obtained we briefly point out how to construct a corres ponding multiresolution analysis and wavelets. (C) 1995 Academic Press , Inc.