Selection theorems, based on generalized variation and oscillation
type: Item
instance of: scholarly article
title: Selection theorems, based on generalized variation and oscillation
zbMATH Open document ID: 0651.26010
DOI: 10.1007/BF02843906
publication date: 1986-01-01T00:00:00Z
review text: Theorems on the existence of convergent subsequences of a sequence of real functions, based on generalized variation and oscillation are proved. A notion of uniformly limited oscillation of real functions, a bit complicated to be reproduced here, is introduced. Among others the following theorem is proved for functions defined on a general set S. Let \(\{y_ i\}\) be a pointwise bounded sequence of real functions on a set S. If \(\{y_ k-y_{\ell}:\quad 1\leq \ell \leq k\}\) is of uniformly limited oscillation, then \(\{y_ i\}\) contains a subsequence which converges pointwise on S. In case the functions are defined on a subset of R, results closely connected with Helly's selection theorem are obtained. Also a new proof of Ascoli's theorem is given.
Mathematics Subject Classification ID: 26A45
zbMATH DE Number: 4061570
zbMATH Keywords: lambda-convergence
zbMATH Keywords: pointwise convergence
zbMATH Keywords: generalized variation
zbMATH Keywords: uniformly limited oscillation
zbMATH Keywords: Helly's selection theorem
zbMATH Keywords: Ascoli's theorem
author: Luisa Di Piazza
author: Caterina Maniscalco
MaRDI profile type: Publication
cites work: Q3686935
cites work: Functions of generalized variation
cites work: A generalization of the Helly selection theorem
cites work: Functions of Φ-Bounded Variation and Riemann-Stieltjes Integration
cites work: On convergence of Fourier series of functions of generalized bounded variation
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4061570