In mathematics, Birkhoff interpolation is an extension of polynomial interpolation.
It refers to the problem of finding a polynomial
such that only certain derivatives have specified values at specified points: where the data points
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.
[1] 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
On the other hand, the Birkhoff interpolation problem where the values of
[2] An important problem in the theory of Birkhoff interpolation is to classify those problems that have a unique solution.
Schoenberg[3] formulates the problem as follows.
denote the number of conditions (as above) and let
be the number of interpolation points.
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?
interpolation points was tackled by George Pólya in 1931.
denote the sum of the entries in the first
columns of the incidence matrix: Then the Birkhoff interpolation problem with
Schoenberg showed that this is a necessary condition for all values of
Consider a differentiable function
Let us see that there is no Birkhoff interpolation quadratic polynomial such that
The derivative of the interpolation polynomial is given by
Consider a differentiable function
Let us see that there is indeed Birkhoff interpolation quadratic polynomial such that
Construct the interpolating polynomial of
is the Birkhoff interpolating polynomial.
Construct the Lagrange/Newton polynomial (same interpolating polynomial, different form to calculate and express them)
is the Birkhoff interpolating polynomial satisfying the above conditions.
is the Birkhoff interpolating polynomial.