Green's function-based wavelets

Abstract

We propose a new approach for the construction of wavelets based on Green's functions associated with boundary value problems. We first describe the rational behind this idea, and then present our first successful results concerning the scaling function and the wavelet associated with Laplace operator. We obtain closed form solutions for the mother wavelet and the scaling function derived from the Laplace operator. We have identified several properties of our wavelet and the associated scaling function: Each of these functions corresponds to a charge-neutral sequence of parallel lines. Our wavelet corresponds to the potential distribution of three equidistant lines with the charge densities 1,-2, and 1. Our scaling function corresponds to the element factor of an infinite array. Furthermore, we will demonstrate the existence of an infinite family of wavelet-like orthogonal systems associated with the Laplace operator. This fact guarantees great flexibilty in constructing appropriate symmetric and asymmetric bases functions. Finally we proceed to the Helmholz wave equation, and construct the scaling function and the wavelet associated with the Helmholz operator. Many of the calculations can be carried out in closed form, allowing to discuss several fascinating properties of these new functions.