Almost everywhere convergence of spline sequences.
Sprache des Titels:
We prove the analogue of the Martingale Convergence Theorem for polynomial spline sequences. Given a natural number k and a sequence (ti) of knots in [0, 1] with multiplicity ? k ? 1, we let Pn be the orthogonal projection onto the space of spline polynomials in [0, 1] of degree k ? 1 corresponding to the grid (ti)ni=1. Let X be a Banach space with the Radon?Nikodým property. Let (gn) be a bounded sequence in the Bochner?Lebesgue space L1X
[0, 1] satisfying
We prove the existence of limn??gn (t) in X for almost every t ? [0, 1]. Already in the scalar valued case X = ? the result is new.