On Mean Convergence of Lagrange Interpolation for General Arrays

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

“…In [2] we proved: Theorem 1. Let K & R be compact, and let v 2 L q ðKÞ for some q > 0: Let the array X of interpolation points lie in K: The following are equivalent:…”

“…The essential feature is that a single condition, namely (2), is sufficient for mean convergence of Lagrange interpolation in L p for at least one p > 0: This should be compared to results surveyed in [3,5,6,9], where amongst other things, the interpolation points are assumed to be zeros of orthogonal polynomials associated with weights satisfying a number of conditions. The price one pays for the simplicity of (2) is that invariably p51 or even p5 1 2 ; and p and r are different in (I) and (II).…”

“…In a recent paper [2], the author used distribution functions to treat general interpolation arrays contained in a compact set. Here we consider the non-compact case, and use decreasing rearrangements of functions, as well as a well-known inequality of Hardy and Littlewood.…”

On Mean Convergence of Trigonometric Interpolants, and Their Unit Circle Analogues, for General Arrays

“…In a recent paper [5], we showed how distribution functions and Loomis' Lemma can be used to obtain a simple necessary and sufficient condition for the existence of ρ > 0 for which Lagrange interpolation polynomials converge in L p . The interest in this lies in the simplicity of the proof and its general applicability.…”

“…The main ideas are similar to those in [5], but there are some technical complications in the proofs. First, however, let us recall the result of [5]. Let X be an array of interpolation points X = {a ; jn} 1 < J < nn>1 in a compact set Κ C R, with…”