Convolution Equations in Colombeau’s Spaces

Lecture Notes: Non-Standard Approach to J.F. Colombeauâ•Žs Theory of Generalized Functions

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physi­ cists and engineers who do research on J. F. Colombeau’ theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau’ theory is the improvement of the properties of the scalars of the varieties of spaces...

Convolution and Homogeneous Spaces

Shift-Invariant Spaces and Linear Operator Equations

In this paper we investigate the structure of finitely generated shift-invariant spaces and solvability of linear operator equations. Fourier transforms and semi-convolutions are used to characterize shift-invariant spaces. Criteria are provided for solvability of linear operator equations, including linear partial difference equations and discrete convolution equations. The results are then ap...

Hypoelliptic Dunkl equations in the space of distributions on R

In this paper we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on R. By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.

4 M ay 2 00 0 MULTILINEAR WEIGHTED CONVOLUTION OF L 2 FUNCTIONS , AND APPLICATIONS TO NON - LINEAR DISPERSIVE EQUATIONS

The X s,b spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behaviour of non-linear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in...