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Czechoslovak Mathematical Journal

, Volume 62, Issue 1, pp 77–104 | Cite as

Integrals and Banach spaces for finite order distributions

  • Erik TalvilaEmail author
Article

Abstract

Let B c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A c n to be the space of tempered distributions that are the nth distributional derivative of a unique function in B c . Similarly with A r n from B r . A type of integral is defined on distributions in A c n and A r n . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A c n and A r n are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B c and B r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A c 1 is the completion of the L 1 functions in the Alexiewicz norm. The space A r 1 contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.

Keywords

regulated function regulated primitive integral Banach space Banach lattice Banach algebra Schwartz distribution generalized function distributional Denjoy integral continuous primitive integral Henstock-Kurzweil integral primitive 

MSC 2010

26A39 46B42 46E15 46F10 46G12 46J10 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.University of the Fraser ValleyAbbotsfordCanada

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