The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts

Abstract

The “classical” wavelets, those ψ eL2(R) such that {2j/2ψ(2jx−k)}, j,keZ, is an orthonormal basis for L2 (R), are known to be characterized by two simple equations satisfied by\(\hat \psi \). The “multiresolution analysis” wavelets (briefly, the MRA wavelets) have a simple characterization and so do the scaling functions that produce these wavelets. Only certain smooth classes of the low pass filters that are determined by these scaling functions, however, appear to be characterized in the literature (see Chapter 7 of [3] for an account of these matters). In this paper we present a complete characterization of all these filters. This somewhat technical result does provide a method for simple constructions of low pass filters whose only smoothness assumption is a Holder condition at the origin. We also obtain a characterization of all scaling sets and, in particular, a description of all bounded scaling sets as well as a detailed description of the class of scaling functions.