On Summability, Multipliability, and Integrability

Seppo Heikkilä () and Antonín Slavík ()
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Seppo Heikkilä: University of Oulu, Department of Mathematical Sciences
Antonín Slavík: Charles University in Prague, Faculty of Mathematics and Physics

A chapter in Current Topics in Summability Theory and Applications, 2016, pp 181-239 from Springer

Abstract: Abstract We define and study summability and multipliability of families indexed by well-ordered sets of real numbers. These concepts generalize the classical notions of convergence of infinite series and products. The members of the families are assumed to be elements of general Banach spaces or Banach algebras, but most of our results are new even in the real-valued case. Our studies are also motivated by problems in integration theory of functions of one variable. In particular, we describe the relation between integrability and product integrability on one side, and summability and multipliability on the other side. Applications in the theory of differential equations with impulses and distributional differential equations are presented, and concrete examples are introduced to illustrate the derived theoretical results.

Keywords: Summability; Multipliability; Well-ordered set; Transfinite induction; Banach space; Banach algebra; Integral; Product integral; Step mapping; Right regulated mapping; Distribution; Differential equation (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-0913-6_5

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DOI: 10.1007/978-981-10-0913-6_5

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