The sharpness of Timan's theorem on differentiable functions
type: Item
instance of: scholarly article
title: The sharpness of Timan's theorem on differentiable functions
zbMATH Open document ID: 0574.41011
DOI: 10.1016/0021-9045(82)90008-9
author: Maurice Hasson
published in: Journal of Approximation Theory
publication date: 1982-01-01T00:00:00Z
review text: Timan's theorem gives sufficient conditions for a function to be in \(C^ 1[a,b]\) in terms of the rate of approximation by polynomials with respect to the modulus of continuity of the function. This paper constructs a \(non\)-C\({}^ 1\) function using Chebyshev polynomials such that the degree of approximation \(E_ n(f)=O(1/(na_ n))\) with \(\sum 1/(na_ n)=\infty\) showing that Timan's condition is sharp. It is also shown that an optimal sufficient condition for f to be in \(C^ 1[a,b]\) is \(\sum nE_ n(f)<\infty\).
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 3916905
zbMATH Keywords: Timan's theorem
zbMATH Keywords: rate of approximation
zbMATH Keywords: Chebyshev polynomials
MaRDI profile type: Publication
full work available at URL: 0021-9045(82)90008-9
OpenAlex ID: W1976376677
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description: scientific article; zbMATH DE number 3916905