Continuous wavelet transform with the Shannon wavelet from the point of view of hyperbolic partial differential equations

TL;DR

This paper presents a novel interpretation of the continuous wavelet transform as the difference of solutions to hyperbolic PDEs, linking wavelet parameters to PDE characteristics and providing a new analytical perspective.

Contribution

It introduces a new representation of the continuous wavelet transform using hyperbolic PDEs, connecting wavelet analysis with PDE theory and offering insights into the transform's internal structure.

Findings

Transform values inside the characteristic triangle are uniquely determined by boundary values and derivatives.

Wavelet shift and scale are treated as independent variables in a PDE framework.

The approach offers a new analytical perspective on wavelet transforms through hyperbolic PDEs.

Abstract

We identify the result of the continuous wavelet transform with the difference of solutions of two hyperbolic partial differential equations, for which wavelet's shift and scale are considered as independent variables on 2D plane. The characteristic property, which follows from the introduced representation is is the fact that the transform's values inside the triangle defined by two characteristics (a=const, b=const) and crossecting them slopped line on a scale-shift plane (a,b) are completely and uniquely defined by the value of transform and its derivatives along the last mentioned line.