Decomposition of spaces of distributions induced by tensor product bases

TL;DR

This paper develops a framework for decomposing spaces of distributions using tensor product bases and needlets, enabling advanced analysis of weighted function spaces on multidimensional domains.

Contribution

It introduces new constructions of multivariate cutoff functions and kernels for tensor product Jacobi polynomials, facilitating the study of weighted Triebel-Lizorkin and Besov spaces.

Findings

Construction of rapidly decaying kernels and frames (needlets) for tensor product Jacobi polynomials.

Development of weighted Triebel-Lizorkin and Besov spaces on multidimensional domains.

Application of cross product bases to distributions on products of complex domains.

Abstract

Rapidly decaying kernels and frames (needlets) in the context of tensor product Jacobi polynomials are developed based on several constructions of multivariate $C^\infty$ cutoff functions. These tools are further employed to the development of the theory of weighted Triebel-Lizorkin and Besov spaces on $[-1, 1]^d$. It is also shown how kernels induced by cross product bases can be constructed and utilized for the development of weighted spaces of distributions on products of multidimensional ball, cube, sphere or other domains.