IJPAM: Volume 35, No. 2 (2007)

A DECOMPOSITION ALGORITHM FOR
FUNCTIONS OF BOUNDED VARIATION

Ibraheem Alolyan
College of Science
King Saud University
P.O. Box 2455, Riyadh, 11451, KINGDOM OF SAUDI ARABIA
e-mail: alolyan@math.colostate.edu


Abstract.The well known Jordan Decomposition Theorem gives the useful characterization that any function of bounded variation can be written as the difference of two increasing functions. Functions which can be expressed in this way can be used to formulate an exclusion test for the recent cellular exclusion algorithms for numerically computing all zero points or the global minima of functions in a given cellular domain. In this paper we give an algorithm to approximate such increasing functions when only the values of the function of bounded variation can be computed. For this purpose, we are led to introduce the idea of $\epsilon$-increasing functions, i.e., functions $f$ such that $f(x)\leq f(y) + \epsilon$ for all $x < y$ in the domain of the function. It is shown that for any Lipschitz continuous function that has finite number of oscillation points, we can find two $\epsilon$-increasing functions such that the Lipschitz function can be written as the difference of these functions.

Received: December 26, 2006

AMS Subject Classification: 26A45, 26A48

Key Words and Phrases: bounded variation, Jordan decomposition, $\epsilon$-increasing, oscillation, oscillation points

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 35
Issue: 2