In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem finding a polynomial p of degree d such that

p^{(n_i)}(x_i) = y_i \qquad\mbox{for } i=1,\ldots,d,

where the data points (x_i,y_i) and the nonnegative integers n_i are given. It differs from Hermite interpolation in that it is possible to specify derivatives of p 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 .

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 p such that p(−1) = p(1) = 0 and p′(0) = 1. On the other hand, the Birkhoff interpolation problem where the values of p′(−1), p(0) and p′(1) 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. formulates the problem as follows. Let d denote the number of conditions (as above) and let k be the number of interpolation points. Given an d-by-k matrix E, all of whose entries are either 0 or 1, such that exactly d entries are 1. Then the corresponding problem is to determine p such that

p^{(j)}(x_i) = y_{i,j} \qquad\text{for all } (i,j) \text{ with } e_{ij} = 1.

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

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}.

The question now becomes: does a Birkhoff interpolation problem with a given incidence matrix have a unique solution for any choice of the interpolation points?

The case with k = 2 interpolation points was tackled by . Let S_m denote the sum of the entries in the first m columns of the incidence matrix:

S_m = \sum_{i=1}^k \sum_{j=1}^m e_{ij}.

Then the Birkhoff interpolation problem with k = 2 has a unique solution if and only if S_m m for all m. showed that this is a necessary condition for all values of k.

References

• .
• .
• .
• .

eo:Interpolo de Birkhoff it:Interpolazione di Birkhoff