Wavelet analysis on the Cantor dyadic group

Compactly supported orthogonal wavelets are built on the Canter dyadic group (the dyadic or 2-series local field). Necessary and sufficient conditions are given on a trigonometric polynomial scaling filter for a multiresolution analysis to result. A Lipschitz regularity condition is defined and an unconditional L-P-convergence result is given for regular wavelet expansions (p > 1). Wavelets are given whose scaling filter is a trigonometric polynomial with 2(n) many terms; regular wavelets with filters with 8 terms are detailed. These wavelets are identified with certain Walsh series on the real line. A Mallat tree algorithm is given for the wavelets.