The Schur l1 Theorem for filters
Antonio Avil\'es, Bernardo Cascales, Vladimir Kadets, Alexander Leonov
TL;DR
This paper investigates classes of filters on natural numbers where weak and strong filter convergence in l1 coincide, and explores an analogue of the weak sequential completeness theorem for filter convergence.
Contribution
It introduces new classes of filters ensuring equivalence of weak and strong convergence in l1 and extends the weak sequential completeness theorem to filter convergence.
Findings
Identified classes of filters with coinciding weak and strong convergence in l1.
Extended the weak sequential completeness theorem to filter convergence.
Provided conditions under which filter convergence behaves similarly to classical convergence.
Abstract
We study the classes of filters F on N such that the weak and strong F-convergence of sequences in l1 coincide. We study also an analogue of l1 weak sequential completeness theorem for filter convergence.
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Taxonomy
TopicsApproximation Theory and Sequence Spaces · Mathematical Analysis and Transform Methods · Digital Filter Design and Implementation