On the sequence of errors in best polynomial approximation (Q1356646)

A classic theorem of Bernstein shows that for any sequence of numbers \(\{a_i\}\) decreasing to zero, there is an \(f\in C[-1,1]\) such that \(E_n(f)=a_n\), where \(E_n(f)\) is the distance from \(f\) to the polynomials of degree not exceeding \(n\). This paper shows that properties of the sequence \(\{a_i\}\) are reflected in properties of the functions, \(f\), that satisfy the Bernstein theorem. General results are stated. More specifically, these are used to answer a question raised by Gaier by showing that for some sequences \(\{a_i\}\), Bernstein's theorem can not be satisfied by an \(f\) that is regular in subsets of [-1,1].