# IJPAM: Volume 35, No. 2 (2007)

**A DECOMPOSITION ALGORITHM FOR**

FUNCTIONS OF BOUNDED VARIATION

FUNCTIONS OF BOUNDED VARIATION

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
-increasing functions, i.e., functions such that
for all 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
-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, -increasing, oscillation, oscillation points

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 35

**Issue:** 2