2011/10/17 by Erik Talvila, Talvila, Erik
Mathematics · #26A39 #46B42 #46E15 #46F10 #46G12 #46J10 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #math.CA #math.FA #msc:26A39 #msc:46B42 #msc:46E15 #msc:46F10 #msc:46G12 #msc:46J10
paper · pdf · doi:10.48550/arxiv.1110.3715
To appear in Czechoslovak Mathematical Journal
arxiv created 2011/10/17 · openalex publication_date 2011/10/17 · arxiv updated 2011/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Bc denote the real-valued functions continuous on the extended real line and vanishing at -∞. Let \Br denote the functions that are left continuous, have a right limit at each point and vanish at -∞. Define \acn to be the space of tempered distributions that are the nth distributional derivative of a unique function in \Bc. Similarly with \arn from \Br. A type of integral is defined on distributions in \acn and \arn. The multipliers are iterated integrals of functions of bounded variation. For each n∈\N, the spaces \acn and \arn are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to \Bc and \Br, 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 \ac1 is the completion of the L1 functions in the Alexiewicz norm. The space \ar1 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.