An analogue to the Edgeworth-Esseen expansion
type: Item
instance of: scholarly article
title: An analogue to the Edgeworth-Esseen expansion
zbMATH Open document ID: 0919.60049
author: Michael R. Snell
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-08-31T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 1217032
zbMATH Keywords: JFM 36.0305.01
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5928
OpenAlex ID: W1990920322
cites work: Norms of powers of absolutely convergent Fourier series
cites work: Q5576559
cites work: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law
cites work: Q4327561
cites work: Q4444545
DOI: 10.1006/JMAA.1998.5928
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description: scientific article; zbMATH DE number 1217032