Construction Of Compactly Supported Symmetric Scaling Functions

Abstract

. In this paper we study scaling functions of a given regularity for arbitrary dilation factor q. We classify symmetric scaling functions and study the smoothness of some of them. We also introduce a new class of continuous symmetric scaling functions, the "Batman" functions, that have very small support. Their smoothness is established. 1. Introduction Compactly supported wavelet functions are typically constructed from multiresolution analyses whose scaling functions are compactly supported, see [6] or [15]. It is an important problem to construct scaling functions (and hence wavelets) that possess desirable properties. These properties usually include high regularity, symmetry and small supports. Recall that a multiresolution analysis with dilation factor q, where q 2 Z and jqj ? 1, is a sequence of nested subspaces of L 2 (R) \Delta \Delta \Delta ae V \Gamma2 ae V \Gamma1 ae V 0 ae V 1 ae V 2 ae \Delta \Delta \Delta (1.1) such that V j = span ae f(q j x \Gamma k) : k 2 Z ..