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Journal of Function Spaces and Applications
Volume 2012, Article ID 202987, 9 pages
http://dx.doi.org/10.1155/2012/202987
Research Article

On Functions of Bounded (p,k)-Variation

1Universidad Central de Venezuela, Caracas, Venezuela
2Universidad Nacional Abierta, Caracas, Venezuela

Received 7 April 2010; Accepted 24 November 2010

Academic Editor: Jürgen Appell

Copyright © 2012 N. Merentes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study the concept of (p,k)-variation (1<p<, kN) of a real function on a compact interval. In particular, we prove that a function u:[a,b]R has bounded (p,k)-variation if and only if u(k-1) is absolutely continuous on [a,b] and u(k) belongs to Lp[a,b]. Moreover, an explicit connection between the (p,k)-variation of u and the Lp-norm of u(k) is given which is parallel to the classical Riesz formula characterizing functions in the spaces RVp[a,b] and Ap[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev space Wpk[a,b].