Spaces with Almost Strictly Totally Positive Bases

In spline spaces there are often totally positive bases possessing a strong property called almost strictly total positivity. In this paper, it is proved that, for totally positive bases of continuous functions B, the following concepts are equivalent: (i) B is almost strictly totally positive, (ii) B satifies a Schoenberg-Whitney Theorem, (iii) The functions in B are locally linearly independent. Some classical examples of almost strictly totally positive bases are given, completing the knowledge of their properties known in the mathematical literature. Some criteria to know the existence of almost strictly totally positive bases are also derived.