On wavelet fundamental solutions to the heat equation -- heatlets (Q1975460)

The application of wavelet theory in differential equations has been mostly focused on numerical computation. The present paper treats an application of wavelet theory to the theory of differential equations. Consider the initial value problem for the heat equation \[ u_t={\sigma^2\over 2}u_{xx}, \quad -\infty<x <\infty,\;t>0 \] with initial data \(u(x,0)= \varphi(x)\). When the initial function \(\varphi(x)\) is a wavelet function, then the solution of the problem is called a heatlet. The paper gives a fundamental property of heatlet, and using this property the authors give a heatlet decomposition theorem of the solution to the initial value problem for the heat equation (heat evolution). As an example the authors investigate the Haar heatlet derived from the Haar multiresolution. The authors treat also the case of multidimensional heat equation, and construct multidimensional heatlets, give the result about the self-similarity and other fundamental properties. Moreover they give the \(n-D\) heatlet decomposition to the heat evolution of dimension \(n\).