The existence in T-spaces of functions with prescribed alternations

AbstractA (k + 1)-dimensional real vector space U of real-valued functions defined on a subset T of the real line is a Tchebycheff space (the linear space generated by a Tchebycheff system) iff the number of zeros and the number of alternations in sign of each nonzero element of U is at most k. We prove here that a necessary and sufficient condition that U be a Tchebycheff space is that for any n ⩽ k (not necessarily distinct) points in T, there exists an element of U with exactly these points as zeros (except for possibly k − n additional zeros), which alternates in sign across each zero. Furthermore, it is proved that if U is a Tchebycheff space of bounded functions, then the prescribed zeros can include points in the closure of T, if for these points “is equal to zero at” is understood to mean “is asymptotically zero at.