Almost everywhere convergence of spline sequences

Martingale convergence Theorems for Tensor Splines

“…The implication (a) =⇒ (b) in item (iv) is also proved in [5], whereas the reverse implication (b) =⇒ (a) is shown in [7] by constructing a non-convergent kmartingale spline sequence with values in Banach spaces X without RNP for any positive integer k.…”

“…We restrict ourselves to this special setting in this section to present the crucial arguments in a concise form. In order to lift the subsequent result from this hypothesis, we use technical arguments in the spirit of those in the proof of the onedimensional result [5,Sections 4 and 6]. This will be presented in detail in Section 5.…”

“…For each ℓ ≤ s, the B-splines (N ℓ n,r ) r whose supports intersect V ℓ j ℓ can be indexed in such a way that for each fixed r, the function N ℓ n,r ½ V ℓ j ℓ converges uniformly to a function Nℓ r as n → ∞ (cf. [5,Section 4]). This is the case since the interior of V ℓ j ℓ does not contain any accumulation points of ∪ n ∆ ℓ n .…”

“…By a standard argument for passing from a weak type (1,1) inequality of a maximal function to a.e. convergence for L 1 X -functions, item (iii) is proved in [9] in the case that ∪ n F n generates the Borel-σ-algebra on I and in [5] in general. We also identify the limit of P k n f as…”

Multivariate orthogonal spline systems

“…(1) L 1 -bounded k-martingale spline sequences (X n ) converge almost everywhere to some L 1 -function. [14,9] (2) Inequality (1.1) holds for k-martingale spline sequences (X n ) with a constant C p,k depending only on p and k but not on the interval filtration (F n ) (see [6] for k = 2 and [12] for general k).…”