In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Birkhoff interpolation is an extension of
polynomial interpolation. It refers to the problem of finding a polynomial
of degree
such that ''only certain''
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s 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
George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body ...
, 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
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...
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