An error term and uniqueness for Hermite–Birkhoff interpolation involving only function values and/or first derivative values

This paper discusses various aspects of Hermite–Birkhoff interpolation that involve prescribed values of a function and/or its first derivative. An algorithm is given that finds the unique polynomial satisfying the given conditions if it exists. A mean value type error term is developed which illustrates the ill-conditioning present when trying to find a solution to a problem that is close to a problem that does not have a unique solution. The interpolants we consider and the associated error term may be useful in the development of continuous approximations for ordinary differential equations that allow asymptotically correct defect control. Expressions in the algorithm are also useful in determining whether certain specific types of problems have unique solutions. This is useful, for example, in strategies involving approximations to solutions of boundary value problems by collocation.