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 $L_X^1[0,1]$ satisfying $$g_n=P_n(g_{n+1}),n\in \mathbb{N}\mathrm{.}$$ We prove the existence of ${\lim }_{n\to \mathrm{\infty }}{g}_n(t)$ in $X$ for almost every $t\in [0,1]\mathrm{.}$ Already in the scalar valued case $X = \mathbb R $ the result is new.