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

Abstract

Let X be a triangular array of interpolation points in a compact subset of [0,2π]. We obtain a necessary and sufficient condition for the existence of ρ > 0 such that the associated trigonometric polynomials are convergent in L p . We also examine Lagrange interpolation on the unit circle. The results are analogues of our earlier ones for Lagrange interpolation on a real interval. 1991 Mathematics Subject classification: 41A05 The ResultIn 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. Most positive results on mean convergence of Lagrange interpolation are closely linked to zeros of orthogonal polynomials, and are somewhat technical -see [6], [8], [12], [13]. An extension to interpolation associated with weights on the real line was presented in [7], using decreasing rearrangements and an inequality of Hardy and Littlewood.In this paper, we shall present an analogue for trigonometric interpolation and for interpolation on the unit circle. 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, withWe denote by L n \ • ] the associated Lagrange interpolation operator, so that for / : Κ R, we have η 3=1 where the fundamental polynomials {4n}* =1 satisfy 4n i x jn) = 5jk.We also let π" denote a polynomial of degree η (without any specific normalisation) whose zeros are {χ 7 · η }" =1 · Our result was:Brought to you by |