A pseudo-probabilistic approach to the dilation equation for wavelets

TL;DR

This paper introduces a pseudo-probabilistic framework for solving the dilation equation in wavelet theory, utilizing probabilistic and number theoretic methods to analyze solutions in one and two dimensions.

Contribution

It develops a novel pseudo-probabilistic approach and variants of the cascade algorithm for the dilation equation, extending previous results in wavelet analysis.

Findings

Existence and absolute continuity of solutions in 1D and 2D

Efficient recovery of classical results in specific cases

Development of adapted cascade algorithms

Abstract

The dilation equation arises naturally when using a multiresolution analysis to construct a wavelet basis. We consider solutions in the space of signed measures, which, after normalization, can be viewed as pseudo-probability measures. Using probabilistic ideas as well as a notion of binary expansions, we discuss the existence and absolute continuity of solutions in dimensions one and two, efficiently recovering previous results in two natural cases. A central role is played by the development of two variants of the classical cascade algorithm, which are adapted to the pseudo-probabilistic and elementary number theoretic context.