Operator-Like Wavelet Bases of $L_2(\mathbb{R}^d)$

TL;DR

This paper introduces a new class of wavelet bases that mimic Fourier multiplier operators, enabling sparse representations of signals modeled as distributions affected by differential operators.

Contribution

It generalizes the connection between derivative operators and wavelets by constructing operator-like wavelet bases from a stochastic model involving white noise and differential operators.

Findings

Wavelet bases inherit sparsity from white noise models.

Construction is fully general and based on underlying operators.

Properties of wavelets are characterized in relation to the operators.

Abstract

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.