Coconvex Approximation of Periodic Functions

D. Leviatan*, I. A. Shevchuk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let C~ be the space of continuous 2 π-periodic functions f, endowed with the uniform norm ‖ f‖ : = max xR| 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.

Original languageEnglish
Pages (from-to)695-726
Number of pages32
JournalConstructive Approximation
Volume57
Issue number2
DOIs
StatePublished - Apr 2023

Funding

FundersFunder number
National Research Foundation of Ukraine2020.02/0155

    Keywords

    • Coconvex approximation by trigonometric polynomials
    • Degree of approximation
    • Jackson-type estimates

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