Almost everywhere convergence of spline sequences
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ALMOST EVERYWHERE CONVERGENCE OF SPLINE SEQUENCES
BY
¨ller and Markus Passenbrunner Paul F. X. Mu Institute of Analysis, Johannes Kepler University Linz Altenberger Strasse 69, 4040 Linz, Austria e-mail: [email protected], [email protected]
ABSTRACT
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 )n i=1 . Let X be a Banach space with the Radon–Nikod´ ym property. Let (gn ) be a bounded sequence in the Bochner–Lebesgue space L1X [0, 1] satisfying gn = Pn (gn+1 ),
n ∈ N.
We prove the existence of limn→∞ gn (t) in X for almost every t ∈ [0, 1]. Already in the scalar valued case X = R the result is new.
1. Introduction In this paper we prove a convergence theorem for splines in vector valued L1 -spaces. By way of introduction we consider the analogous convergence theorems for martingales with respect to a filtered probability space (Ω, (An ), μ). We first review two classical theorems for scalar valued martingales in L1 = L1 (Ω, μ). See Neveu [6].
Received November 7, 2017 and in revised form August 20, 2019
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¨ P. F. X. MULLER AND M. PASSENBRUNNER
Isr. J. Math.
(M1) Let g ∈ L1 . If gn = E(g|An ) then gn 1 ≤ g1 and (gn ) converges almost everywhere and in L1 . (M2) Let (gn ) be a bounded sequence in L1 such that gn = E(gn+1 |An ). Then (gn ) converges almost everywhere and g = lim gn satisfies g1 ≤ sup gn 1 . Next we turn to vector valued martingales. We fix a Banach space X and let L1X = L1X (Ω, μ) denote the Bochner–Lebesgue space. The Radon–Nikod´ ym property (RNP) of the Banach space X is intimately tied to martingales in Banach spaces. We refer to the book by Diestel and Uhl [3] for the following basic and well known results. (M3) Let g ∈ L1X . If gn = E(g|An ) then gn L1X ≤ gL1X . The sequence (gn ) converges almost everywhere in X and in L1X . (This holds for any Banach space X.) (M4) Let (gn ) be a bounded sequence in L1X such that gn = E(gn+1 |An ). If the Banach space X satisfies the Radon–Nikod´ ym property, then (gn ) converges almost everywhere in X and g = lim gn satisfies gL1X ≤ sup gn L1X . Moreover, the L1X -density of the μ-absolutely continuous part of the vector measure ν(E) = lim
n→∞
gn dμ,
E ∈ ∪An
E
determines g = lim gn . (M5) Conversely, if X fails to satisfy the Radon–Nikod´ ym property, then there exists a filtered probability space (Ω, (An ), μ) and bounded sequence in L1X (Ω, μ) satisfying gn = E(gn+1 |An ) such that (gn ) fails to converge almost everywhere in X. In the present paper we establish a new link between probability (almost sure convergence of martingales, the RNP) and approximation theory (projections onto splines in [0, 1]). We review the basic setting pertaining to spline projections. (See, for instance, [12], [9], [11].) So, fix an integer k ≥ 2, and let (ti ) be a sequen
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