Dilation equations and the smoothness of compactly supported wavelets

Abstract

In this chapter, the authors characterize all wavelets arising from finite-coefficient dilation equations which are smooth and compactly supported- such wavelets are necessarily well-localized in the time domain and have good decay in their Fourier transforms. Daubechies and Lagarias independently discovered the specific implementation, and gave sufficient conditions for the existence of smooth, compactly supported scaling functions and lower bounds for the corresonding Holder exponents of the last derivative in terms of a joint spectral radius. In general, the relationship between the smoothness of scaling functions and the coefficients of the dilation equation is much more complicated than is apparent from the discussion of four-coefficient dilation equations. Daubechies has shown that, for real-valued coefficients, symmetry precludes the possibility of an associated multiresolution analysis, with the single exception of the Haar system. Thus these scaling functions cannot be used to construct wavelet orthonormal bases.