The Uniform Convergence Ordinal Index and the l1-Behavior of a Sequence of Functions

The Uniform Convergence Ordinal Index and the l1-Behavior of a Sequence of Functions Positivity 8: 49–74, 2004. 49 © 2004 Kluwer Academic Publishers. Printed in the Netherlands. The Uniform Convergence Ordinal Index and the l -Behavior of a Sequence of Functions VASSILIKI FARMAKI Department of Mathematics, University of Athens, Panepistimiopolis, 6-15784 Athens, Greece. E-mail: vfarmaki@math.uoa.gr (Received 21 March 2001; accepted 1 July 2002) Abstract. In this paper we introduce and study two indices of a uniformly bounded sequence f of real valued functions deﬁned on a set  and converging pointwise to a function f . The ﬁrst index f −f f  n measures uniform convergence of f , while the second index  measures the relation n + of the sequence f −f to the positive face of the usual basis of  . There is a close connection between these two indices, indicated by: f −f f  n (a)  < ⇔ < ; and 1 + 1 f −f n n (b) if  < then  = where is the least ordinal with   . 1 + Using this connection the following dichotomies hold: f  1 either [Case  = f −f has an l -subsequence; 1 n or [Case  < f  http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

The Uniform Convergence Ordinal Index and the l1-Behavior of a Sequence of Functions

, Volume 8 (1) – Oct 21, 2004
26 pages

/lp/springer_journal/the-uniform-convergence-ordinal-index-and-the-l1-behavior-of-a-74zrFar4Sq
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/B:POST.0000023197.60454.3b
Publisher site
See Article on Publisher Site

Abstract

Positivity 8: 49–74, 2004. 49 © 2004 Kluwer Academic Publishers. Printed in the Netherlands. The Uniform Convergence Ordinal Index and the l -Behavior of a Sequence of Functions VASSILIKI FARMAKI Department of Mathematics, University of Athens, Panepistimiopolis, 6-15784 Athens, Greece. E-mail: vfarmaki@math.uoa.gr (Received 21 March 2001; accepted 1 July 2002) Abstract. In this paper we introduce and study two indices of a uniformly bounded sequence f of real valued functions deﬁned on a set  and converging pointwise to a function f . The ﬁrst index f −f f  n measures uniform convergence of f , while the second index  measures the relation n + of the sequence f −f to the positive face of the usual basis of  . There is a close connection between these two indices, indicated by: f −f f  n (a)  < ⇔ < ; and 1 + 1 f −f n n (b) if  < then  = where is the least ordinal with   . 1 + Using this connection the following dichotomies hold: f  1 either [Case  = f −f has an l -subsequence; 1 n or [Case  < f 

Journal

PositivitySpringer Journals

Published: Oct 21, 2004

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