Constrained Approximation with Jacobi Weights

Kopotun, Kirill ; Leviatan, Dany ; Shevchuk, Igor

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 109-128

Résumé

In this paper, we prove that for $\ell \,=\,1$ or 2 the rate of best $\ell $ - monotone polynomial approximation in the ${{L}_{p}}$ norm $\left( 1\,\le \,p\,\le \,\infty\right)$ weighted by the Jacobi weight ${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with $\alpha ,\,\beta \,>\,-1/p$ if $p\,<\,\infty $ , or $\alpha ,\,\beta \,\ge \,0$ if $p\,=\,\infty $ , is bounded by an appropriate $\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

Détail

DOI : 10.4153/CJM-2015-034-4

Mots-clés : 41A29, 41A10, 41A15, 41A17, 41A25, constrained approximation, Jacobi weights, weighted moduli of smoothness, exact estimates, exact orders

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