On Bi-dimensional Second Variation

• Autorzy: Ereu J. , Gimémez J. , Merentes N.
• Języki publikacji: EN

Abstrakty

EN

In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in R2. We use Hardy-Vitali type technics in the plane in order to extend the classical notion of function of bounded second variation on intervals of R. We introduce the class [formula] of all functions of bounded second variation on a rectangle [formula] and show that this class can be equipped with a norm with respect to which it is a Banach space. Finally, we present two results that show that integrals of functions of first bounded variation are in [formula].

Słowa kluczowe

EN

functions of bounded second variation; functions of bounded variation

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2012
• Tom: Vol. 52, [Z] 1
• Strony: 39--59
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Ereu J.

autor

Gimémez J.

autor

Merentes N.

• Departamento de Matematicas, Decanato de Ciencias y Tecnologia Universidad Centroccidental Lisandro Alvarado Barquisimeto, Venezuela,

Bibliografia

• [1] R. Adams and J. A. Clarkson, Properties of functions f(x; y) of Bounded Variation, Trans. Amer. Math. Soc. 36 (1934), 711-730.
• [2] V. V. Chistyakov, Superposition Operators in the Algebra of Functions of two Variables with Finite Total Variation, Monatshefte fur Mathematik 137 (2002), 99-114.
• [3] P. L. Dirichlet., Sur la convergence des sćries trigonemetriques que servent d representer une function arbitraire entre des limites donnes, Journal fur die Reine und Angewandte Mathematik 4 (1826), 157-159.
• [4] G. H. Hardy, On double fourier series, and especially those which represent the double zeta-function width real and inconmesurable parameters, Quart. J. Math. Oxford. 37 (1905/06), 53-79.
• [5] T. H. Hildebrandt, Introduction to the theory of integration, Academic Press , New York, 1963.
• [6] C. Jordan, sur la serie de fourier, C. R. Acad. Sci. Paris 2 (1881), 228-230.
• [7] A. B. Owen Multidimensional variation for quasi-Monte Carlo, Stanford University, 2004.
• [8] N. Merentes, S. Rivas, El Operador de Composición en Espacios de Funciones con algun tipo de Variacion Acotada, IX Escuela Venezolana de Matematicas, Facultad de Ciencias-ULA, Merida- Venezuela, 1996.
• [9] F. Riesz: Sur certains systems singvliers d'equations integrates, Annales de L'Ecole Norm. Sup., Paris 3 28 (1911), 33-68.
• [10] A. W. Roberts and D. E. Varberg: Functions of bounded convexity, Bull. Amer. Math. Soc. Volume 75, Number 3 (1969), 568-572.
• [11] A. W. Roberts and D. E. Varberg: Convex Functions, Academic Press, New York-London, 1973.
• [12] A. M. Russell and C. J. F. Upton, A generalization of a theorem by F. Riesz, Analysis Mathematica 9 (1983), 69-77.
• [13] F. A. Talalyan, A multidimensional analogue of a theorem of F. Riesz, Sb. Math. Volume 186, Number 9 (1995), 1363-1374.
• [14] Ch. J. de la Valle Poussin, Sur la convergence des formules d'interpolation entre ordennćes equidistantes, Bull. Accad. Set. Belg; (1908), 314-410.
• [15] G. Vitali, Sulle funzioni integral!, Atti Accad. Schi. Torino CI Sci. Fis. Mat. Natur. 40 (1904/05), 1021-1034.
• [16] M. Wróbel, Of functions of bounded n-th variation, Annales Mathematicae Silesianae 15 (2001), 79-86.