One sided Hermite interpolation by piecewise different generalized polynomials

Piecewise generalized polynomials of different kinds of order n (ECT-splines of order n) are constructed from different ECT-systems of order n via connection matrices which are nonsingular and totally positive.A well-known zero count for polynomial splines is extended to ECT-splines. It is used to construct ECT-B-splines and to show under which conditions ECT-splines will solve modified Hermite-type interpolation problems. Also conditions are specified such that piecewise generalized polynomials form rECT-systems and the interpolation problems associated with may be solved recursively.