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# Constrained Approximation with Jacobi Weights

Published:2015-10-23
Printed: Feb 2016
• Kirill Kopotun,
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba
• Dany Leviatan,
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
• Igor Shevchuk,
Faculty of Mechanics and Mathematics, National Taras Shevchenko University of Kyiv, Kyiv, Ukraine
 Format: LaTeX MathJax PDF

## Abstract

In this paper, we prove that, for $\ell=1$ or $2$, the rate of best $\ell$-monotone polynomial approximation in the $L_p$ norm ($1\leq p \leq \infty$) weighted by the Jacobi weight $w_{\alpha,\beta}(x) :=(1+x)^\alpha(1-x)^\beta$ with $\alpha,\beta\gt -1/p$ if $p\lt \infty$, or $\alpha,\beta\geq 0$ if $p=\infty$, is bounded by an appropriate $(\ell+1)$st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $(\ell+2)$nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.
 Keywords: constrained approximation, Jacobi weights, weighted moduli of smoothness, exact estimates, exact orders
 MSC Classifications: 41A29 - Approximation with constraints 41A10 - Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 41A15 - Spline approximation 41A17 - Inequalities in approximation (Bernstein, Jackson, Nikol'skiii-type inequalities) 41A25 - Rate of convergence, degree of approximation

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