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February 2021 Sample numbers and optimal Lagrange interpolation in Sobolev spaces
Guiqiao Xu, Hui Wang
Rocky Mountain J. Math. 51(1): 347-361 (February 2021). DOI: 10.1216/rmj.2021.51.347

Abstract

This paper investigates the optimal recovery of Sobolev spaces Wr[1,1],r in space L[1,1] and weighted spaces Lp,ω[1,1],1p< with ω a continuous integrable weight function in (1,1). We obtain the values of the sampling numbers of Wr[1,1] in L[1,1] and Lp,ω[1,1],1p<. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for p=. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree r with the leading coefficient 1 of the least deviation from zero in Lp,ω[1,1] are optimal for 1p<. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.

Citation

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Guiqiao Xu. Hui Wang. "Sample numbers and optimal Lagrange interpolation in Sobolev spaces." Rocky Mountain J. Math. 51 (1) 347 - 361, February 2021. https://doi.org/10.1216/rmj.2021.51.347

Information

Received: 14 January 2020; Accepted: 6 April 2020; Published: February 2021
First available in Project Euclid: 28 May 2021

Digital Object Identifier: 10.1216/rmj.2021.51.347

Subjects:
Primary: 41A05 , 41A25 , 41A46
Secondary: 65D05

Keywords: optimal Lagrange interpolation algorithm , sampling numbers , Sobolev space

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 1 • February 2021
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