On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

Lubinsky, D. S.

doi:10.1006/jath.2002.3698

Cited by 6 publications

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“…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.…”

Section: The Resultsmentioning

confidence: 98%

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

Lubinsky¹

2002

Analysis

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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 |

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Section: The Resultsmentioning

confidence: 98%

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

Lubinsky¹

2002

Analysis

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“…The problem of "distributing well" a large number of points on certain D ≥ 1-dimensional compact sets embedded in R D+1 is an interesting problem with numerous wide applications in diverse areas for example approximation theory, zeroes of extremal polynomials in all kinds of settings, singular operators for example Hilbert transforms, random matrix theory, crystal and molecule structure, electrostatics, special functions, Newtonian energy, extensions, alignment, number theory, manifold learning, clustering, shortest paths, codes and discrepancy, vision, signal processing and many others. See for example some references in [7,97,98,14,15,35,36,37,38,28,29,30,32,45,33,34,49,95,96,23,50,89,31,47,19,50,51,53,100,79,43] for a good overview. For some of our work in various approximation processes, operator theory, extremal polynomials, combinatorial and t-designs, codes and finite fields, extremal configurations, energy, discrepancy, manifold learning, potential theory, shortest paths, signal processing, vision and extensions can be found for example in the papers [6,7,22,23,24,28,29,31,…”

Section: Main Resultmentioning

confidence: 99%

On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

Damelin¹

2021

Preprint

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This paper is an exposition of work of the author et al. detailing fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic-differential geometry, analysis on manifolds, complex-harmonic analysis, data science, partial differential equations, optimization and probability. A significant portion of the work is based on joint research with Charles Fefferman in the papers [39,40,41,42]. The topics of this work include (a) The space of maps of bounded mean oscillation (BMO) in R D , D ≥ 2. (b) The labeled and unlabeled near alignment and Procrustes problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (c) The Whitney near isometry extension problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (d) Partitions and clustering of compact sets and point sets with certain geometries in R D , D ≥ 2 and analysis on certain manifolds in R D , D ≥ 2. Many open problems for future research are given.

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A tribute to Géza Freud

Mhaskar

2004

Journal of Approximation Theory

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On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

Cited by 6 publications

References 6 publications

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

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

On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

A tribute to Géza Freud

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