Almost everywhere Convergence of Spline Sequences

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 $\le k-1$, 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-Nikod\'{y}m property. Let $(g_n)$ be a bounded sequence in the Bochner-Lebesgue space $L^1_X [0,1]$ satisfying $$g_n = P_n ( g_{n+1} ),\qquad n \in \mathbb N .$$ 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 $X = \mathbb R$ the result is new.