Martingale convergence theorems for tensor splines

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–Nikodým property of Banach spaces X in terms of pointwise X-valued martingale convergence are obtained in this setting. Those assertions are in full analogy with their martingale counterparts and hold independently of filtration, spline degree, and dimension d.