Coconvex approximation of periodic functions

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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“…g − T n , the error of the best coconvex approximation of the function g. It is known [10] that if f ∈ ∆ (2) , then…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Negative results in coconvex approximation of periodic functions

Dzyubenko

Voloshyna

Yushchenko

2021

Journal of Approximation Theory

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“…The fact that P n is a polynomial from T cn can be directly verified arithmetically like in [14], or [3], using (4.5), i.e., all the arithmetical terms in (4.5), having been evaluated in the sum (4) together with the corresponding divided differences, including the L 3 , are equal 0. Verify (1.2).…”

Section: Construction Of the Nearly Coconvex Polynomialmentioning

confidence: 99%

“…In papers [10] and [14] two coconvex analogues of the inequality (1.1) were proved for k = 2 and k = 3, respectively. Moreover, in [15] arguments from the papers [11], [12] of Shvedov and [1] of DeVore, Leviatan and Shevchuk were used to show that for k > 3 there is no coconvex analogue of the inequality (1.1).…”

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confidence: 99%

Jackson's type estimate of nearly coconvex approximation

Dzyubenko

2016

Preprint

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Suppose that a continuous on the real axis 2π-periodic function f changes its convexity at 2s, s ∈ N, points y i on each period: −π ≤ y 2s < y 2s−1 < ... < y 1 < π, and for the rest i ∈ Z, the points y i are defined periodically. In the paper, for each n ≥ N, a trigonometric polynomial P n of order cn is found such that: P n has the same convexity as f, everywhere except, perhaps, the small neighborhoods of the y i :(where N is a constant depending only on min i=1,...,2s {y i − y i+1 }, c and c(s) are constants depending only on s, ω 4 (f, •) is the modulus of continuity of the 4-th order of the function f, and • is the max-norm.

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