Article Dans Une Revue Journal of Approximation Theory Année : 2021

NEGATIVE RESULTS IN COCONVEX APPROXIMATION OF PERIODIC FUNCTIONS

Résumé

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 that (1) f (t) 2s i=1 (t − yi) ≥ 0, t ∈ [y2s, y0], and for each trigonometric polynomial Tn of degree ≤ n (of order ≤ 2n + 1), satisfying (2) T n (t) 2s i=1 (t − yi) ≥ 0, t ∈ [y2s, y0], the inequality n r−1 f − Tn C(R) ≥ cr f (r) 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} 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 sup n→∞ n r f − Tn C(R) ω4(f (r) , 1/n) = +∞, where ω4 is the fourth modulus of continuity.

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hal-02956219 , version 1 (02-10-2020)

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German Dzyubenko, Victoria Voloshina, Lyudmyla Yushchenko. NEGATIVE RESULTS IN COCONVEX APPROXIMATION OF PERIODIC FUNCTIONS. Journal of Approximation Theory, 2021, ⟨10.1016/j.jat.2021.105582⟩. ⟨hal-02956219⟩

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