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Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.48 no.2 Bogotá July/Dec. 2014 


A Generalization for the Riesz p-Variation

Una generalización de la p-variación de Riesz


1Universidad Nacional de Colombia, Bogotá, Colombia. Email:
2Pontificia Universidad Javeriana, Bogotá, Colombia. Email:
3Universidad de Oriente, Cumaná, Venezuela. Email:


In this paper we introduce a generalization of the concept of Riesz p-variation and construct a function space which is normalizable and moreover is a Banach space as well as a Banach algebra. Furthermore, using Medved'ev approach we obtain an integral characterization of the functions in this function space.

Key words: Riesz p-variation, (\phi, α)-Bounded variation, Bounded variation.

2000 Mathematics Subject Classification: 26A45, 26B30, 26A16, 26A24.


En este artículo se introduce una generalización del concepto de \hboxp-variación de Riesz y se construye un espacio de funciones que es normalizable y además es tanto espacio de Banach como un álgebra de Banach. Adicionalmente, usando el enfoque dado por Medved'ev se obtiene una caracterización integral de las funciones en dicho espacio funcional.

Palabras clave: p-Variación de Riesz, variación (\phi, α)-acotada, variación acotada.

Texto completo disponible en PDF


[1] J. Appell, J. Banás, and N. J. Merentes, Bounded Variation and Around, de Gruyter, Berlin, Germany,         [ Links ] 2014.

[2] R. E. Castillo, H. Rafeiro, and E. Trousselot, 'Embeddings on Spaces of Generalized Bounded Variation', Rev. Colomb. Mat. 48, 1 (2014), 97-109.         [ Links ]

[3] R. E. Castillo and E. Trousselot, 'On Functions of (p,α)-Bounded Variation', Real Anal. Exchange 34, 1 (2009), 49-60.         [ Links ]

[4] M. C. Chakrabarty, 'Some Results On \mathsfAC-ω Functions', Fund. Math. 64, (1969a), 219-230.         [ Links ]

[5] M. C. Chakrabarty, 'Some Results on ω-Derivatives and \mathsfBV-ω Functions', J. Austral. Math. Soc. 9, (1969b), 345-360.         [ Links ]

[6] P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, USA,         [ Links ] 1950.

[7] R. L. Jeffery, 'Generalized Integrals with respect to Functions of Bounded Variation', Canad. J. Math. 10, (1958), 617-626.         [ Links ]

[8] C. Jordan, 'Sur la série de Fourier', C. R. Acad. Paris 2, (1881), 228-230.         [ Links ]

[9] L. Maligranda and W. Orlicz, 'On Some Properties of Functions of Generalized Variation', Monatsh. Math. 104, 1 (1987), 53-65.         [ Links ]

[10] Y. T. Medved'ev, 'Generalization of a Theorem of F. Riesz', Uspehi Matem. Nauk (N.S.) 8, 6(58) (1953), 115-118.         [ Links ]

[11] J. Musielak, Orlicz Spaces and Modular Spaces, Vol. 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany,         [ Links ] 1983.

[12] B. G. Pachpatte, Mathematical Inequalities, North-Holland Mathematical Library, Elsevier Science,         [ Links ] 2005.

[13] F. Riesz, 'Untersuchungen über Systeme Integrierbarer Funktionen', Math. Ann. 69, 4 (1910), 449-497.         [ Links ]

[14] A. I. Vol'pert and S. I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Vol. 8 of Mechanics: Analysis, Martinus Nijhoff Publishers, Dordrecht, Holland,         [ Links ] 1985.

(Recibido en enero de 2014. Aceptado en septiembre de 2014)

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

    AUTHOR  = {Castillo, René Erlin and Rafeiro, Humberto and Trousselot, Eduard},
    TITLE   = {{A Generalization for the Riesz \boldsymbol{p}-Variation}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2014},
    volume  = {48},
    number  = {2},
    pages   = {165--190}