Hermite–Birkhoff interpolation by Hermite–Birkhoff splines

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E . Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F . We first find necessary conditions on ( E , F ) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on ( E , F ) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.