Almost everywhere Convergence of Spline Sequences

2017/11/06 by Müller, Paul F. X., Passenbrunner, Markus
#42C10 #46B22 #65D07 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/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 ≤ 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=1ⁿ. Let X be a Banach space with the Radon-Nikodým property. Let (g_n) be a bounded sequence in the Bochner-Lebesgue space L¹_X [0,1] satisfying g_n = P_n ( g_n+1 ), n ∈ \mathbb N . We prove the existence of lim_{n→ ∞} g_n(t) in X for almost every t ∈ [0,1]. Already in the scalar valued case X = \mathbb R the result is new.

Related

• Martingale convergence Theorems for Tensor Splines
• Spline Characterizations of the Radon-Nikodým property
• Martingale inequalities for spline sequences
• Orthogonal projectors onto spaces of periodic splines
• On almost everywhere convergence of tensor product spline projections