Birkhoff interpolation
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial of degree such that only certain derivatives have specified values at specified points: where the data points and the nonnegative integers are given. It differs from Hermite interpolation in that it is possible to specify derivatives of at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff, who first studied the problem in 1906. Existence and uniqueness of solutions In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution. For instance, there is no quadratic polynomial such that and . On the other hand, the Birkhoff interpolation problem where the values of and are given always has a unique solution. An important problem in the theory of Birkhoff interpolation is to classify those problems that have a unique solution. Schoenberg formulates the problem as follows. Let denote the number of conditions (as above) and let be the number of interpolation points. Given a matrix , all of whose entries are either or , such that exactly entries are , then the corresponding problem