Total positivity and the existence of piecewise exponential B-splines

We give necessary and sufficient conditions for the total positivity of certain connection matrices arising in piecewise exponential spline spaces. These total positivity conditions are sufficient for existence of B-splines in such spaces, but they are far from being necessary. We give a necessary and sufficient condition for existence of B-splines in the case of piecewise exponential spline spaces with only two differential operators, which eventually leads to a necessary condition for any piecewise exponential spline spaces. Dedicated to Professor Charles A. Micchelli for his 60th birthday Mathematics subject classifications (2000) 65D17, 65D07, 41A15, 41A50.