New estimates related with the best polynomial approximation

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DOI:

https://doi.org/10.33993/jnaat521-1313

Keywords:

Favard inequality, best approximation
Abstract views: 131

Abstract

In some old results, we find estimates the best approximation \(E_{n,p}(f)\) of a periodic function satisfying \(f^{(r)}\in\mathbb{L}^p_{2\pi}\) in terms of the norm of \(f^{(r)}\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \(f^{(r)}\in \mathbb{L}^q_{2\pi}\), with \(1<q<p<\infty\). We will present inequalities of the form \(E_{n,p}(f)\leq C(n)\Vert D^{(r)}f\Vert_q\), where \(D^{(r)}\) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.

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References

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G. Gasper, Positivity and the convolution structure for Jacobi series, Ann. of Math., 93 (1971), pp. 112-118. https://doi.org/10.2307/1970755 DOI: https://doi.org/10.2307/1970755

G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Annals of Mathematics, 95 (2) (1972), pp. 261-280. https://doi.org/10.2307/1970800 DOI: https://doi.org/10.2307/1970800

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Published

2023-07-10

How to Cite

Bustamante, J. (2023). New estimates related with the best polynomial approximation. J. Numer. Anal. Approx. Theory, 52(1), 75–81. https://doi.org/10.33993/jnaat521-1313

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