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Search: MSC category 41A29 ( Approximation with constraints )

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1. CJM Online first

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 orders
Categories:41A29, 41A10, 41A15, 41A17, 41A25

2. CJM 2005 (vol 57 pp. 1224)

Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A.
Convex Polynomial Approximation in the Uniform Norm: Conclusion
Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian-Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Categories:41A10, 41A25, 41A29

3. 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 space
Categories:41A10, 41A15, 41A25, 41A29

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