The sharpness of Timan's theorem on differentiable functions
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 \(\sum nE_ n(f)<\infty\).
scientific article; zbMATH DE number 3916905