Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Kopotun, Kirill A.; Leviatan, D.; Shevchuk, I. A.

doi:10.1007/s11253-010-0362-2

Cited by 9 publications

(7 citation statements)

References 9 publications

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“…The case q = 2, 2 < α, N = s + 3 of Theorem 2 resolves the question posed in ( [5], see comments for the table s ≥ 3, pp. 424-425) in that we have "⊕" for these (α, N ).…”

Section: )supporting

confidence: 53%

“…This overview deals only with comonotone approximation, q = 1; the complete picture is known also for q = 2, see [4,5] and Corollary 3, while for larger q the full generality has not been achieved. For a survey of shape-preserving, in particular q-coconvex, approximation see [3].…”

Section: Overview Of Comonotone Approximationmentioning

confidence: 99%

“…Under the above hypotheses, [5,Theorem 3.3] is applicable, so ω ϕ s+1,q ( f (q) , t) ≤ c(α, s, q)t α−q ; in particular, f (q) exists in (−1, 1) and is continuous there. Using [2, Chapter 7, Theorem 3.4] or [9, Theorem 18.1], we obtain that there exists a polynomial W ∈ P s+q+1 as desired.…”

Section: )mentioning

confidence: 99%

“…Note that the general approach from [5][6][7][8] etc., used in comonotone and coconvex approximation to establish results of the type described in Definition 1, is inapplicable here. For instance, usage of weighted moduli in estimating the degree of approximation is impossible directly, as can be seen from counterexamples [ …”

Section: )mentioning

confidence: 99%

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On the degree of piecewise shape-preserving approximation by polynomials

Vlasiuk¹

2015

Journal of Approximation Theory

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We show that the degree of piecewise q-coconvex approximation E (q) n ( f, Y s ) of a q-convex function by algebraic polynomials of degree < n asymptotically depends only on α, Y s and q,assuming this inequality holds for an N * dependent on f and that the degree of unconstrained approximation has the same decrease rate: n α E n ( f ) ≤ 1, n ≥ N with N ≤ s + q + 1. In particular, cases α > 1, N = s + 2, q = 1, and α > 2, N = s + 3, q = 2 complement earlier results on comonotone and coconvex approximation by polynomials respectively.

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“…The case q = 2, 2 < α, N = s + 3 of Theorem 2 resolves the question posed in ( [5], see comments for the table s ≥ 3, pp. 424-425) in that we have "⊕" for these (α, N ).…”

Section: )supporting

confidence: 53%

Section: Overview Of Comonotone Approximationmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

On the degree of piecewise shape-preserving approximation by polynomials

Vlasiuk¹

2015

Journal of Approximation Theory

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“…This implies, in particular, that these moduli are the right measure of smoothness to be used while investigating constrained weighted approximation (see e.g. [3,7,8]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Moduli of Smoothness with Jacobi Weights

2018

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The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α (1 + x) β for functions in the Jacobi weighted L p [−1, 1], 0 < p ≤ ∞, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted L p spaces. If 1 ≤ p ≤ ∞, then these moduli are equivalent to certain weighted K-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these p),

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New Moduli of Smoothness: Weighted DT Moduli Revisited and Applied

2014

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We introduce new moduli of smoothness for functions f ∈ L p [−1, 1] ∩ C r −1 (−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that ϕ r f (r ) These moduli are equivalent to certain weighted Ditzian-Totik (DT) moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in L p [−1, 1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.Communicated by Kamen Ivanov.Part of this work was done while the first two authors were at

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Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Cited by 9 publications

References 9 publications

On the degree of piecewise shape-preserving approximation by polynomials

On the degree of piecewise shape-preserving approximation by polynomials

On the Moduli of Smoothness with Jacobi Weights

New Moduli of Smoothness: Weighted DT Moduli Revisited and Applied

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