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Search: MSC category 41A15 ( Spline approximation )

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1. CJM 2015 (vol 68 pp. 109)

Kopotun, Kirill; Leviatan, Dany; Shevchuk, Igor
 Constrained Approximation with Jacobi Weights 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 ordersCategories:41A29, 41A10, 41A15, 41A17, 41A25

2. CJM 1997 (vol 49 pp. 74)

Hu, Y. K.; Kopotun, K. A.; Yu, X. M.
 Constrained approximation in Sobolev spaces Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions $f\in\Wp^k\mll$ are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained. Keywords:Constrained approximation, polynomials, splines, degree of, approximation, $L_p$ space, Sobolev spaceCategories:41A10, 41A15, 41A25, 41A29
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