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| In [[mathematics]], '''Birkhoff interpolation''' is an extension of [[polynomial interpolation]]. It refers to the problem finding a polynomial ''p'' of degree ''d'' such that certain [[derivative]]s have specified values at specified points:
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| :<math> p^{(n_i)}(x_i) = y_i \qquad\mbox{for } i=1,\ldots,d, </math>
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| where the data points <math>(x_i,y_i)</math> and the nonnegative integers <math>n_i</math> 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 {{harvtxt|Birkhoff|1906}}.
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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 ''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 {{harv|Passow|1983}}.
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| An important problem in the theory of Birkhoff interpolation is to classify those problems that have a unique solution. {{harvtxt|Schoenberg|1966}} formulates the problem as follows. Let ''d'' denote the number of conditions (as above) and let ''k'' be the number of interpolation points. Given a ''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
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| :<math> p^{(j)}(x_i) = y_{i,j} \qquad\text{for all } (i,j) \text{ with } e_{ij} = 1. </math> | |
| The matrix ''E'' is called the incidence matrix. For example, the incidence matrices for the interpolation problems mentioned in the previous paragraph are:
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| :<math> \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}. </math>
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| 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?
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| The case with ''k'' = 2 interpolation points was tackled by {{harvtxt|Pólya|1931}}. Let ''S<sub>m</sub>'' denote the sum of the entries in the first ''m'' columns of the incidence matrix:
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| :<math> S_m = \sum_{i=1}^k \sum_{j=1}^m e_{ij}. </math>
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| Then the Birkhoff interpolation problem with ''k'' = 2 has a unique solution if and only if ''S<sub>m</sub>'' ≥ ''m'' for all ''m''. {{harvtxt|Schoenberg|1966}} showed that this is a necessary condition for all values of ''k''.
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| ==References==
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| * {{Citation | last1=Birkhoff | first1=George David | author1-link=George David Birkhoff | title=General mean value and remainder theorems with applications to mechanical differentiation and quadrature | jstor=1986339 | year=1906 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=7 | pages=107–136 | issue=1 | publisher=American Mathematical Society}}.
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| * {{Citation | last1=Passow | first1=Eli | title=Book Review: Birkhoff interpolation by G. G. Lorentz, K. Jetter and S. D. Riemenschneider | doi=10.1090/S0273-0979-1983-15204-7 | year=1983 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=9 | issue=3 | pages=348–351}}.
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| * {{Citation | doi=10.1002/zamm.19310110620 | last1=Pólya | first1=George | author1-link=George Pólya | title=Bemerkung zur Interpolation und zur Naherungstheorie der Balkenbiegung | year=1931 | journal=[[Journal of Applied Mathematics and Mechanics]] | issn=0044-2267 | volume=11 | pages=445–449}}.
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| * {{Citation | last1=Schoenberg | first1=Isaac Jacob | author1-link=Isaac Jacob Schoenberg | title=On Hermite-Birkhoff interpolation | doi=10.1016/0022-247X(66)90160-0 | year=1966 | journal=Journal of Mathematical Analysis and Applications | issn=0022-247X | volume=16 | pages=538–543}}.
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| [[Category:Interpolation]]
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