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.[1]

Existence and uniqueness of solutions

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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.[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. 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.[4] 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

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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

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  1. ^ Birkhoff, George David (1906). "General mean value and remainder theorems with applications to mechanical differentiation and quadrature". Transactions of the American Mathematical Society. 7 (1): 107–136. doi:10.1090/S0002-9947-1906-1500736-1. ISSN 0002-9947.
  2. ^ "American Mathematical Society". American Mathematical Society. Retrieved 2022-05-19.
  3. ^ Schoenberg, I. J (1966-12-01). "On Hermite-Birkhoff interpolation". Journal of Mathematical Analysis and Applications. 16 (3): 538–543. doi:10.1016/0022-247X(66)90160-0. ISSN 0022-247X.
  4. ^ Pólya, G. (1931). "Bemerkung zur Interpolation und zur Näherungstheorie der Balkenbiegung". ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik (in German). 11 (6): 445–449. doi:10.1002/zamm.19310110620.