An analogue to the Edgeworth-Esseen expansion
Let \(f\) be a bounded, measurable function on \([-\pi, \pi]\) and let \((a_{n\nu}: \nu=0,\pm 1,\pm 2,\dots)\) be the Fourier coefficients of the powers \(f^n\) for \(n= 1,2,\dots\)\ . The asymptotic behaviour of \(a_{n\nu}\) is studied in the case where (i) \(\sup\{| f(t)|:\varepsilon<| t|<\pi\}< 1\) for every \(\varepsilon>0\), (ii) \(\log f(t)= i\alpha t+ At^q+ o(t^q)\) as \(t\to 0\), (iii) \(q\) is even and \(\text{Re }A\neq 0\). The asymptotic expansion of \(a_{n\nu}\) obtained holds uniformly in \(\nu\) as \(n\to\infty\). There is a significant distinction between the cases \(q= 2\) and \(q\geq 4\). In the special case \(a_{\nu 0}\geq 0\) for all \(\nu\), the same asymptotic expansion was first obtained by \textit{F. Y. Edgeworth} [Trans. Cambridge Philos. Soc. 20, 36-65 and 113-141 (1905; JFM 36.0305.01)].
scientific article; zbMATH DE number 1217032