Editing Birkhoff interpolation (section)

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In [[mathematics]], '''Birkhoff interpolation''' is an extension of [[polynomial interpolation]]. It refers to the problem of finding a polynomial <math>P(x)</math> of degree <math>d</math> such that ''only certain'' [[derivative]]s have specified values at specified points:
:<math> P^{(n_i)}(x_i) = y_i \qquad\mbox{for } i=1,\ldots,d, </math>
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 <math>P(x)</math> 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.<ref>{{Cite journal |last=Birkhoff |first=George David |date=1906 |title=General mean value and remainder theorems with applications to mechanical differentiation and quadrature |url=https://www.ams.org/tran/1906-007-01/S0002-9947-1906-1500736-1/ |journal=Transactions of the American Mathematical Society |language=en |volume=7 |issue=1 |pages=107–136 |doi=10.1090/S0002-9947-1906-1500736-1 |issn=0002-9947|doi-access=free }}</ref>