The representation of a wavelet transform of the Gaussian family by a superposition of solutions to partial differential equations

Abstract

The usage of partial differential equations for the evaluation of a wavelet transform with real and complex wavelets and with vanishing higher moments is considered. Contrary to the case of the transform with the standard Morlet wavelet, the sought-for transform can be found as a superposition of solutions to several Cauchy problems with various initial values. These initial values are the products of a transformed function with some power functions whose exponents vary from zero to the maximal number of a vanishing moment.