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Martingale convergence theorems for tensor splines

Volume 274 / 2024

Markus Passenbrunner Studia Mathematica 274 (2024), 11-36 MSC: Primary 41A15; Secondary 42B25, 46B22, 42C10, 60G48 DOI: 10.4064/sm220925-10-11 Published online: 15 January 2024

Abstract

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 $d$.

Authors

  • Markus PassenbrunnerInstitute of Analysis
    Johannes Kepler University Linz
    4040 Linz, Austria
    e-mail
    e-mail

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