Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

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

doi:10.5802/smai-jcm.54

Cited by 12 publications

(11 citation statements)

References 31 publications

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“…In other words, we conjecture, that the truth table of the validity of Jackson type estimates in the coconvex approximation of periodic functions has the same form as the truth table of Jackson type estimates in the coconvex approximation of non-periodic functions by algebraic polynomials, see [6], Page 114, Fig. 3, or [4], Page 62, Table 24. Remark 1.4.…”

Section: Introduction and The Main Resultsmentioning

confidence: 86%

Negative results in coconvex approximation of periodic functions

Dzyubenko

Voloshyna

Yushchenko

2021

Journal of Approximation Theory

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We prove, that for each r ∈ N, n ∈ N and s ∈ N there are a collection {yi} 2s i=1 of points y2s < y2s−1 < • • • < y1 < y2s + 2π =: y0 and a 2π -periodic function f ∈ C (∞) (R), such thatand for each trigonometric polynomial Tn of degree ≤ n (of order ≤ 2n + 1), satisfying(2)holds, where cr > 0 is a constant, depending only on r. Moreover, we prove, that for each r = 0, 1, 2 and any such collection {yi} 2s i=1 there is a 2π -periodic function f ∈ C (r) (R), such that (−1) i−1 f is convex on [yi, yi−1], 1 ≤ i ≤ 2s, and, for each sequence {Tn} ∞ n=0 of trigonometric polynomials Tn, satisfying (2), we have lim supω4(f (r) , 1/n) = +∞, where ω4 is the fourth modulus of continuity.

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Section: Introduction and The Main Resultsmentioning

confidence: 86%

Negative results in coconvex approximation of periodic functions

Dzyubenko

Voloshyna

Yushchenko

2021

Journal of Approximation Theory

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“…Together with the classical inverse theorems (see e.g. [9,Theorem 5 and Corollary 6]), this implies that, if α > 0, then a function f is in Lip * α if and only if (1.3) inf Pn∈Πn ρ −α n (x)(f (x) − P n (x)) = O(1).…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…More discussions of various related results on monotone approximation can be found in our survey paper [10]. and denote by ∆ (1) the class of all nondecreasing functions on [−1, 1], and by Π n the space of algebraic polynomials of degree ≤ n.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Interpolatory pointwise estimates for monotone polynomial approximation

Kopotun

Leviatan

Shevchuk

2018

Journal of Mathematical Analysis and Applications

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Given a nondecreasing function f on [−1, 1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates "interpolatory estimates".In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f, ·) of f evaluated at √ 1 − x 2 /n and were valid for all n ≥ 1. The current paper is devoted to proving that if f ∈ C r [−1, 1], r ≥ 1, then the interpolatory estimates are valid for the second modulus of smoothness of f (r) , however, only for n ≥ N with N = N(f, r), since it is known that such estimates are in general invalid with N independent of f .Given a number α > 0, we write α = r + β where r is a nonnegative integer and 0 < β ≤ 1, and denote by Lip * α the class of all functions f on [−1, 1] such that ω2(f (r) , t) = O(t β ). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α ≥ 2 since 1985:

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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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Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

Cited by 12 publications

References 31 publications

Negative results in coconvex approximation of periodic functions

Negative results in coconvex approximation of periodic functions

Interpolatory pointwise estimates for monotone polynomial approximation

On the Moduli of Smoothness with Jacobi Weights

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