Methods of solving dilation equations †

Abstract: A wavelet basis is an orthonormal basis for L(R), the space of square-integrable functions on the real line, of the form {gnk}n,k∈Z, where gnk(t) = 2 n/2 g(2t − k) and g is a single fixed function, the wavelet. Each multiresolution analysis for L(R) determines such a basis. To find a multiresolution analysis, one can begin with a dilation equation f(t) = ∑ ck f(2t− k). If the solution f (the scaling function) satisfies certain requirements, then a multiresolution analysis and hence a wavelet basis will follow. This paper surveys methods of achieving this goal. Two separate problems are involved: first, solving a general dilation equation to find a scaling function, and second, determining when such a scaling function will generate a multiresolution analysis. We present two methods for solving dilation equations, one based on the use of the Fourier transform and one operating the time domain utilizing linear algebra. The second method characterizes all continuous, integrable scaling functions. We also present methods of determining when a multiresolution analysis will follow from the scaling function. We discuss simple conditions on the coefficients {ck} which are “almost” sufficient to ensure the existence of a wavelet basis, in particular, they do ensure that {gnk}n,k∈Z is a tight frame, and we present more complicated necessary and sufficient conditions for the generation of a multiresolution analysis. The results presented are due mainly to Cohen, Colella, Daubechies, Heil, Lagarias, Lawton, Mallat, and Meyer, although several of the results have been independently investigated by other groups, including Berger, Cavaretta, Dahmen, Deslauriers, Dubuc, Dyn, Eirola, Gregory, Levin, Micchelli, Prautzsch, and Wang.