JOURNAL OF APPROXIMATION THEORY | 卷:267 |
Negative results in coconvex approximation of periodic functions | |
Article | |
Dzyubenko, German1  Voloshyna, Victoria2,3  Yushchenko, Lyudmyla3  | |
[1] NAS Ukraine, Inst Math, UA-01024 Kiev, Ukraine | |
[2] Taras Shevchenko Natl Univ Kyiv, Fac Mech & Math, UA-01601 Kiev, Ukraine | |
[3] Univ Toulon & Var, F-83130 La Garde, France | |
关键词: Shape preserving approximation; Trigonometric polynomial; Jackson; Convex; | |
DOI : 10.1016/j.jat.2021.105582 | |
来源: Elsevier | |
【 摘 要 】
We prove, that for each r is an element of N, n is an element of N and s. N there are a collection {yi}(i=1)(2s) of points y(2s) < y2(s-1) < center dot center dot center dot < y(1) < y(2s) + 2 pi =: y(0) and a 2 pi - periodic function f is an element of C-(infinity)(R), such that f ''(t).2s pi ''(t)pi(2s)(i=1)(t - yi) >= 0, t is an element of [y(2s), y(0]), (1) and for each trigonometric polynomial T-n of degree <= n (of order <= 2n + 1), satisfying T' n (t).2s pi ''(t)pi(2s)(i=1)(t - yi) >= 0, t is an element of [y(2s), y(0]), (1) the inequality nr(-1)parallel to f - Tn parallel to C(R) = cr >= f ((r))parallel to C(R) 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}(i=1)(2s) there is a 2 pi - periodic function f is an element of 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)(infinity) of trigonometric polynomials Tn, satisfying (2), we have limn(->infinity) sup nr parallel to f - T-n parallel to C(R)/w(4)(f((r)), 1/n = +infinity, where omega(4) is the fourth modulus of continuity. (C) 2021 Elsevier Inc. All rights reserved.
【 授权许可】
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