TY - JOUR

T1 - Coconvex Approximation of Periodic Functions

AU - Leviatan, D.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/4

Y1 - 2023/4

N2 - Let C~ be the space of continuous 2 π-periodic functions f, endowed with the uniform norm ‖ f‖ : = max x∈R| f(x) | , and denote by ωk(f, t) , the k-th modulus of smoothness of f. Denote by C~ r, the subspace of r times continuously differentiable functions f∈ C~ , and let Tn, be the set of trigonometric polynomials Tn of degree ≤ n (that is, of order ≤ 2 n+ 1). Given a set Ys:={yi}i=12s, of 2s points, s≥ 1 , such that - π≤ y1< y2< ⋯ < y2s< π, and a function f∈ C~ r, r≥ 3 , that changes convexity exactly at the points Ys, namely, the points Ys are all the inflection points of f. We wish to approximate f by trigonometric polynomials which are coconvex with it, that is, satisfy f′′(x)Tn′′(x)≥0,x∈R.We prove, in particular, that if r≥ 3 , then for every k, s≥ 1 , there exists a sequence {Tn}n=N∞, N= N(r, k, Ys) , of trigonometric polynomials Tn∈ Tn, coconvex with f, such that ‖f-Tn‖≤c(r,k,s)nrωk(f(r),1/n).It is known that one may not take N independent of Ys.

AB - Let C~ be the space of continuous 2 π-periodic functions f, endowed with the uniform norm ‖ f‖ : = max x∈R| f(x) | , and denote by ωk(f, t) , the k-th modulus of smoothness of f. Denote by C~ r, the subspace of r times continuously differentiable functions f∈ C~ , and let Tn, be the set of trigonometric polynomials Tn of degree ≤ n (that is, of order ≤ 2 n+ 1). Given a set Ys:={yi}i=12s, of 2s points, s≥ 1 , such that - π≤ y1< y2< ⋯ < y2s< π, and a function f∈ C~ r, r≥ 3 , that changes convexity exactly at the points Ys, namely, the points Ys are all the inflection points of f. We wish to approximate f by trigonometric polynomials which are coconvex with it, that is, satisfy f′′(x)Tn′′(x)≥0,x∈R.We prove, in particular, that if r≥ 3 , then for every k, s≥ 1 , there exists a sequence {Tn}n=N∞, N= N(r, k, Ys) , of trigonometric polynomials Tn∈ Tn, coconvex with f, such that ‖f-Tn‖≤c(r,k,s)nrωk(f(r),1/n).It is known that one may not take N independent of Ys.

KW - Coconvex approximation by trigonometric polynomials

KW - Degree of approximation

KW - Jackson-type estimates

UR - http://www.scopus.com/inward/record.url?scp=85140843175&partnerID=8YFLogxK

U2 - 10.1007/s00365-022-09597-y

DO - 10.1007/s00365-022-09597-y

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AN - SCOPUS:85140843175

SN - 0176-4276

VL - 57

SP - 695

EP - 726

JO - Constructive Approximation

JF - Constructive Approximation

IS - 2

ER -