  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 $d$.
