Integral representation of functions of bounded second Φ-variation in the sense of Schramm

• Autorzy: Gimenez J. , Merentes N. , Rivas S.
• Języki publikacji: EN

Abstrakty

EN

In this article we introduce the concept of second Φ--variation in the sense of Schramm for normed-space valued functions defined on an interval [a; b] ⊂ R. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of φ-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function [formula] (where X is a reflexive Banach space) is of bounded second Φ-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.

Słowa kluczowe

EN

Young function; phi-variation; second phi-variation of a function

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 1
• Strony: 137--151
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Gimenez J.

autor

Merentes N.

autor

Rivas S.

• Universidad de Los Andes Departamento de Matemáticas Facultad de Ciencias Mérida, Venezuela,

Bibliografia

• [1] V. Barbu, Th. Precupanu, Convexity and Optimization on Banach Spaces, Sijthof and Noordhoff, the Nederlands, 1978.
• [2] M. Bracamonte, J. Gimenez, N. Merentes, On generalized second Φ -variation of normed space valued maps, J. Math. Control Sci. Appl. (JMCSA) 1 (2011) 4, 75–83.
• [3] V.V. Chistyakov, On mappings of bounded variation, J. Dyn. Control Syst. 3 (1997) 2, 261–289.
• [4] V.V. Chistyakov, Mappings of bounded Φ -variation with arbitrary function Φ, J. Dyn. Control Syst. 4 (1998) 2, 217–247.
• [5] J. Ciemnoczlowski, W. Orlicz, Functions of bounded '-variation and some propierties, Demonstratio Math. 18 (1985), 231–251.
• [6] Ch.J. de la Vallée Poussin, Sur la convergence des formules d’interpolation entre ordennées equidistantes, Bull. Acad. Sei. Belg. (1908), 314–410.
• [7] J. Diestel, J.J. Uhl, Vector measures, Math. Surveys, 15, Amer. Math. Soc. 1977.
• [8] P.L. Dirichelt, Sur la convergence des séries trigonemétriques que servent á représenter une function arbitraire entre des limites donnés, J. Reine Angew. Math. 4 (1826), 157–159.
• [9] A. Hernández, S. Rivas, Funciones de Φ -variación acotada en el sentido de Schramm, Ponencia presentada en las IX jornadas de Matemáticas, Maracaibo, Venezuela, 1996.
• [10] C. Jordan, Sur la série de Fourier, C.R. Acad. Sci. Paris 2 (1881), 228–230.
• [11] F. Riesz, Sur certains systems singuliers d’equations integrates, Annales de L’Ecole Norm. Sup., Paris, 3 (1911) 28, 33–68.
• [12] A.M. Russell, C.J.F. Upton, A generalization of a theorem by F. Riesz, Anal. Math. 9 (1983), 69–77.
• [13] M. Schramm, Functions of -bounded variation and Riemann-Stieltjes integration, Trans. Amer. Math. Soc. 267 (1985) 1, 49–63.
• [14] N. Wiener, Sur une généralisation de la notion de variation de pussance piéme au sens de N. Wiener et sur la convergence des séries de Fourier, C.R. Acad. Sci. París, Ser A-B 204 (1937), 470–472.
• [15] L.C. Young, The quadratic variation of function and its Fourier coefficients, J. Mass. Inst. Technology 3 (1924), 73–94.