Abstract view
Constrained Approximation with Jacobi Weights


Published:20151023
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
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
$w_{\alpha,\beta}(x)
:=(1+x)^\alpha(1x)^\beta$ 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.
MSC Classifications: 
41A29, 41A10, 41A15, 41A17, 41A25 show english descriptions
Approximation with constraints Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} Spline approximation Inequalities in approximation (Bernstein, Jackson, Nikol'skiiitype inequalities) Rate of convergence, degree of approximation
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'skiiitype inequalities) 41A25  Rate of convergence, degree of approximation
