Boundedness of minimax series from values of the function and its derivative (Q1362350)

The so-called ``minimax series'' given by the expression \(S(f)= \sum^\infty_{k= 0} E_k(f)\), represents the sum of all the errors \(E_k(f)\), \(k=0,1,2, \dots\), corresponding to the best approximations of a function \(f\) (continuous on an interval \([a,b])\) on the set \(\Pi_k\) of all polynomials of degree \(\leq k\), with the uniform norm. In this paper the author proves that if \(f\) is a continuous function on \([0,1]\) with \(S(f) <\infty\), then the well-known Bernstein polynomials \((B_nf)(x) =\sum^n_{k=0} {n\choose k} f(k/n) x^k(1-x)^{n-k}\) converge to \(f\) in the norm \(S\). As a consequence it follows \(\lim_{n\to\infty} S(B_nf) =S(f)\) and this result is used to establish certain computable and useful bounds of \(S(f)\) which involve only values of \(f\) and its derivative.