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

The "classical" wavelets, those psi is an element of L-2(R) such that {2(j/ 2)psi(2(j)x - k)}, j, k is an element of Z, is an orthonormal basis for L-2 (R), are known to be characterized by two simple equations satisfied by <(p si)over cap>. The "multiresolution analysis" wavelets (briefly, the MRA wav elets) 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 char acterized in the literature (see Chapter 7 of [3] for an account of these m atters). In this paper we present a complete characterization of all these filters. This somewhat technical result does provide a method for simple co nstructions of low pass filters whose only smoothness assumption is a Holde r 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.