Strong approximation by Fourier series (Q1089858)

In this survey paper the author discusses the influence of \textit{G. Freud} on strong approximation of Fourier series. The first question is how fast a given function f can be approximated by the form \(\{| (n+1)^{- \beta}\sum^{n}_{k=o}(k+1)^{\beta -1}| S_ k(x)-f(x)|^ p\}^{1/p},\) where \(S_ k\) denotes the kth partial sum of the Fourier series of f. A result for \(f^{(r)}\in Lip\alpha\) is stated. Converse results deduce properties of the function f from order of strong approximation. The author explains Freud's results in this area and points out the futher developments up to this day.