Pointwise Estimates for Convex Polynomial Approximation

DeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators H ns j which leave the cones of y-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators H ns j, show that they basically have the same shape preservation behavior while interpolating at the endpoints of [-1, 1], and also satisfy Telyakovskil-and DeVoreGopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results

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“…THEOREM D. (Yu [32], Leviatan [22]). If f e C [ -l , 1] is a convex function then there is a convex polynomial P n e U n , such that…”

Section: Theorem B (Beatson [J Theorem 2]) Let J Be a Positive Intmentioning

confidence: 99%

Pointwise estimates for higher order convexity preserving polynomial approximation

Cao

Gonska

1994

J. Aust. Math. Soc. Series B, Appl. Math

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“…The following estimates of the degree of convex polynomial approximation of functions f ∈ B r ∩ ∆ 2 were proved by Leviatan [11] (r = 1 and 2) and by Kopotun [6] (r = 3 and r ≥ 5):…”

Section: Introductionmentioning

confidence: 96%

“…In fact, Leviatan [11] and Kopotun [8] have obtained estimates refining those in (1.1) and involving, respectively, the Ditzian-Totik (D-T) moduli [3], and the weighted D-T moduli of smoothness (see [16]), defined later in this section. In particular, the following result gives a complete answer, in the case of convex approximation, to a central question in approximation theory, namely, to characterize those (convex) functions with prescribed degree of (convex) polynomial approximation.…”

Section: Introductionmentioning

confidence: 99%

Convex Polynomial Approximation in the Uniform Norm: Conclusion

Kopotun¹,

Leviatan²,

Shevchuk³

2005

Can. j. math.

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Abstract. 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 kth 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.

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“…More recently R. DeVore and X. Yu [5] have given a constructive proof of Timan -Teljackovski type pointwise estimates for monotone polynomial approximation involving the second modulus of smoothness ߱ ଶ . Also D. Leviatan [6] presented pointwise estimates involving ߱ ଶ and providing convex polynomial approximation, as well as simultaneous monotone and convex polynomial approximation. In addition, using a suitable Peetre functional, D. Leviatan [7] obtained estimates with respect to ߱ ଶ of the Jackson type on the rate of the monotone polynomial approximation.…”

Section: Introductionmentioning

confidence: 99%