A representation theorem for operators on a space of interval functions

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) ￠ abfHd ±+ ￠ ‘i=1 ￠ [H(xi ￠ ’1,xi ￠ ’1+) ￠ ’H(xi ￠ ’1+,xi ￠ ’1+)] 2(xi ￠ ’1)+ ￠ ‘i=1 ￠ [H(xi ￠ ’,xi) ￠ ’H(xi ￠ ’,xi ￠ ’)] (xi ￠ ’1,xi)where each of ±, 2, and depend only on T, ± is of bounded variation, 2 and 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.