Almost everywhere Convergence of Spline Sequences

Paul F. X. Müller; Markus Passenbrunner

paper

Almost everywhere Convergence of Spline Sequences

arXiv:1711.01859

Abstract

We prove the analogue of the Martingale Convergence Theorem for polynomial spline sequences. Given a natural number $k$ and a sequence $(t_{i})$ of knots in $[0, 1]$ with multiplicity $\leq k - 1$ , we let $P_{n}$ be the orthogonal projection onto the space of spline polynomials in $[0, 1]$ of degree $k - 1$ corresponding to the grid $(t_{i})_{i = 1}^{n}$ . Let $X$ be a Banach space with the Radon-Nikodým property. Let $(g_{n})$ be a bounded sequence in the Bochner-Lebesgue space $L_{X}^{1} [0, 1]$ satisfying $g_{n} = P_{n} (g_{n + 1}), n \in N .$ We prove the existence of $lim_{n \to \infty} g_{n} (t)$ in $X$ for almost every $t \in [0, 1] .$ Already in the scalar valued case $X = R$ the result is new.