Extensions of endpoint equivalent and periodic Tchebycheff systems

Abstract

Let IZ > 0 be an integer. Given an ordered set A of cardinality at least II + 2, a sequence of real functions (yi}r= 0 defined on A will be called a Tchebycheff system (T-system) on A, provided that, for every sequence 0 0, *.., t,} of points from A such that to y1+ 2) when the functions are periodic with period equal to the length of the set A and are a T-system on A, the set A containing either its infimum, II, or its supremum, I,. Similar to periodicity but not totally coincident is the concept of “endpoint equivalence.” A real function f defined on a real, nonempty set A will be called endpoint equivalent provided that, for all sequences (xn}, { y,} in A, such that x, + I, and yn -+ I,, the limits lim f (x,), lim f (y) exist (finite or infinite), and are equal. A T-system defined on such a set A will be called endpoint equivalent if the functions in it are endpoint equivalent. Although the functions considered should be real valued, one can often permit A to be a subset of the extended real number system. The advantage of this will be seen in what follows.