International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 3, Pages 285-296

A representation theorem for operators on a space of interval functions

J. A. Chatfield

Department of Mathematics, Southwest Texas State University, San Marcos 78666, Texas, USA

Received 4 May 1978

Copyright © 1978 J. A. Chatfield. 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.


Suppose N is a Banach space of norm || and R is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of TH where H is a function from R×R to N such that H(p+,p+), H(p,p+), H(p,p), and H(p,p) each exist for each p and T is a bounded linear operator on the space of all such functions H. In particular we show that TH=(I)abfHdα+i=1[H(xi1,xi1+)H(xi1+,xi1+)]β(xi1)+i=1[H(xi,xi)H(xi,xi)]Θ(xi1,xi)where each of α, β, and Θ depend only on T, α is of bounded variation, β and Θ are 0 except at a countable number of points, fH is a function from R to N depending on H and {xi}i=1 denotes the points P in [a,b]. for which [H(p,p+)H(p+,p+)]0 or [H(p,p)H(p,p)]0. We also define an interior interval function integral and give a relationship between it and the standard interval function integral.