Multivariate orthogonal spline systems

TL;DR

This paper investigates the convergence properties of multivariate orthogonal spline systems, demonstrating almost everywhere and unconditional convergence in L^p spaces under specific geometric conditions.

Contribution

It introduces convergence results for tensor product spline systems, including new conditions ensuring unconditional convergence in L^p spaces.

Findings

Proves almost everywhere convergence of the orthogonal spline series.

Establishes unconditional convergence in L^p spaces under geometric partition conditions.

Identifies geometric criteria affecting convergence behavior.

Abstract

In this article we consider orthonormal systems consisting of tensor products of splines. We show some convergence results of the corresponding orthogonal series including a.e. convergence and unconditional convergence in $L^p$ for $1<p<\infty$, where the latter is proved under some geometric conditions on the involved partitions that depend on the spline order.