Deux résultats sur les suites limite-périodiques. (Two results on limit-periodic sequences)
type: Item
instance of: scholarly article
title: Deux résultats sur les suites limite-périodiques. (Two results on limit-periodic sequences)
zbMATH Open document ID: 0622.10039
author: Jean-Loup Mauclaire
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: First the author proves the following result. If f is an integrable function in the sense of Riemann and a(n) a Besicovič-limit-periodic sequence, then the convolution limit \[ \lim_{N\to \infty}(1/N)\sum_{n\leq N}f(n/N) a(n)\quad = \] \[ =\quad (\lim_{N\to \infty}(1/N)\sum_{n\leq N}f(n/N)) (\lim_{N\to \infty}(1/N)\sum_{n\leq N}a(n)) = \int^{1}_{0}f(t)dt \cdot \int_{G}A dm \] exists (for the definition of G, dm and \(A\in {\mathcal L}(G,dm)\) see the author's book ''Intégration et théorie des nombres'' (Paris 1986), generalizing a result of \textit{C. Radoux} [Ann. Soc. Sci. Brux., Sér. I 91, 13-18 (1977; Zbl 0347.41022)], where a(n) was Euler's function. For the proof, the function a is approximated by certain partial sums of its Fourier series. Secondly, the author answers a problem, posed by the reviewer, in a survey lecture at the Banach Center 1982 [see Banach Cent. Publ. 17, 463- 498 (1985; Zbl 0598.10049)] on Hankel determinants.
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 4008507
zbMATH Keywords: almost-periodic arithmetical function
zbMATH Keywords: Riemann-integrable function
zbMATH Keywords: Fourier expansion
zbMATH Keywords: Ramanujan expansion
zbMATH Keywords: Sylvester determinant
zbMATH Keywords: limit- periodic sequence
zbMATH Keywords: Hankel determinants
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
full work available at URL: pjaa.63.90
OpenAlex ID: W2045521487
cites work: Q4116874
cites work: Q3731703
cites work: Q3764226
cites work: Q5520799
DOI: 10.3792/PJAA.63.90
description: scientific article; zbMATH DE number 4008507