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 is to determine  such that

The matrix  is called the incidence matrix. For example, the incidence matrices for the interpolation problems mentioned in the previous paragraph are:

Now the question is: Does a Birkhoff interpolation problem with a given incidence matrix  have a unique solution for any choice of the interpolation points? The case with  interpolation points was tackled by George Pólya in 1931. Let  denote the sum of the entries in the first  columns of the incidence matrix:

Then the Birkhoff interpolation problem with  has a unique solution if and only if . Schoenberg showed that this is a necessary condition for all values of . Some examples

Consider a differentiable function  on , such that . Let us see that there is no Birkhoff interpolation quadratic polynomial such that  where : Since , one may write the polynomial as  (by completing the square) where  are merely the interpolation coefficients. The derivative of the interpolation polynomial is given by . This implies , however this is absurd, since  is not necessarily . The incidence matrix is given by:

Consider a differentiable function  on , and denote  with . Let us see that there is indeed Birkhoff interpolation quadratic polynomial such that  and . Construct the interpolating polynomial of  at the nodes , such that . Thus the polynomial :  is the Birkhoff interpolating polynomial. The incidence matrix is given by:

Given a natural number , and a differentiable function  on , is there a polynomial such that:  and  for  with ? Construct the Lagrange/Newton polynomial (same interpolating polynomial, different form to calculate and express them)  that satisfies  for , then the polynomial  is the Birkhoff interpolating polynomial satisfying the above conditions. The incidence matrix is given by:

Given a natural number , and a  differentiable function  on , is there a polynomial such that:  and  for ? Construct  as the interpolating polynomial of  at  and , such that . Define then the iterates  . Then  is the Birkhoff interpolating polynomial. The incidence matrix is given by:

References
Interpolation