Martingale convergence Theorems for Tensor Splines

TL;DR

This paper establishes martingale convergence theorems for tensor product splines in multi-dimensional Euclidean spaces, extending classical martingale results to spline-based projections with broad applicability.

Contribution

It introduces tensor spline-based martingale convergence theorems, generalizing classical results to higher dimensions and spline settings, independent of filtration, degree, and dimension.

Findings

Proves pointwise convergence theorems for tensor spline martingales.

Establishes versions of Doob's maximal inequality in this context.

Characterizes Banach space properties via spline martingale convergence.

Abstract

In this article we prove martingale type pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space ($d$ is a positive integer), where conditional expectations are replaced by their corresponding tensor spline orthoprojectors. Versions of Doob's maximal inequality, the martingale convergence theorem and the characterization of the Radon-Nikod\'{y}m property of Banach spaces $X$ in terms of pointwise $X$-valued martingale convergence are obtained in this setting. Those assertions are in full analogy to their martingale counterparts and hold independently of filtration, spline degree, and dimension $d$.