On a Characterisation of Matrix Functions which are Differences of Two Monotone Matrix Functions

Abstract

The class of matrix functions of ’bounded variation’ was introduced by O. Dobsch in a paper published in 1937 [2]. The consideration of this class of functions immediately gives rise to the consideration of those matrix functions of order n on an interval [a, b] that are representable as the difference of two monotone matrix functions on that interval. Such a difference will have high regularity properties when n is large and is therefore much more than simply a function of bounded variation. The characterization of this class was sought in the paper of Dobsch [2]. The purpose of this paper is to give a complete description of a related class: the functions defined on (—1,1) which have restrictions to any closed subinterval which are such differences.