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.
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
26A39 46B42 46E15 46F10 46G12 46J10
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V. G. Čelidze, A. G. Džvaršeĭšvili: The Theory of the Denjoy Integral and Some Applications. Transl. from the Russian by P. S. Bullen. World Scientific, Singapore, 1989.Google Scholar
A. G. Das, G. Sahu: An equivalent Denjoy type definition of the generalized Henstock Stieltjes integral. Bull. Inst. Math., Acad. Sin. 30 (2002), 27–49.MathSciNetzbMATHGoogle Scholar
N. Dunford, J. T. Schwartz: Linear Operators. Part I: General theory. With the assistance of William G. Bade and Robert G. Bartle. Repr. of the orig., publ. 1959 by John Wiley & Sons Ltd., Paperback ed. New York etc.: John Wiley & Sons Ltd. xiv, 1988.Google Scholar
R. J. Fleming, J. E. Jamison: Isometries on Banach Spaces: Function spaces. Chapman and Hall, Boca Raton, 2003.zbMATHGoogle Scholar
G. B. Folland: Real Analysis. Modern Techniques and Their Applications. 2nd ed. Wiley, New York, 1999.zbMATHGoogle Scholar
A. H. Zemanian: Distribution Theory and Transform Analysis. An Introduction to Generalized Functions, with Applications. Reprint, slightly corrected. Dover Publications, New York, 1987.zbMATHGoogle Scholar
W. P. Ziemer: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer-Verlag, Berlin, 1989.zbMATHGoogle Scholar