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

Let X be a triangular array of interpolation points in a compact subset of [0; 2 ]. We obtain a necessary and su¢ cient condition for the existence of p > 0 such that the associated trigonometric polynomials are convergent in Lp. We also examine Lagrange interpolation on the unit circle. The results are analogues of our earlier ones for Lagrange interpolation on a real interval. 1. The Result In a recent paper [5], we showed how distribution functions and Loomis’Lemma can be used to obtain a simple necessary and su¢ cient condition for the existence of p > 0 for which Lagrange interpolation polynomials converge in Lp. 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 = fxjng1 j n;n 1 in a compact set K R, with xnn < xn 1;n < < x2n < x1n: We denote by Ln[ ] the associated Lagrange interpolation operator, so that for f : K ! R, we have Ln[f ](x) = n X j=1 f (xjn) `jn(x); where the fundamental polynomials f`kngnk=1 satisfy `kn (xjn) = jk: We also let n denote a polynomial of degree n (without any speci…c normalisation) whose zeros are fxjngnj=1. Our result was: THEOREM 1 Let K R be compact, and let v 2 Lq (K) for some q > 0. Let the array X of Date : 6 December 2001. 1