Formulas for complex interpolation

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Constructing Craig Interpolation Formulas

1 Background and Introduction Let 27 and H be two inconsistent first order theories. Then by Craig's Interpolation Theorem, there is a sentence 8, called a Craig interpolant, such that 8 is t rue in 27 and false in H and every nonlogical symbol occurring in 8 occurs in bo th 27 and H. Craig interpolants can be used to solve the problem of learning a first order concept by letting 27 and H be th...

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Stability of Barycentric Interpolation Formulas for Extrapolation

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...

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On Error Formulas for Multivariate Polynomial Interpolation

In this paper we prove that the existence of an error formula of a form suggested in [2] leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an application, we provide a negative answer to one version of the question posed by Carl de Boor (cf. [2]) regarding the existence of certain minimal error formulas for multivariate interpolation. §

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On Error Formulas for Multivariate Interpolation

In this paper we prove that the existence of an error formula of a form suggested in [2] leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an application, we provide a negative answer to one version of the question posed by Carl de Boor (cf. [2]) regarding the existence of certain minimal error formulas for multivariate interpolation.

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Stability of Barycentric Interpolation Formulas

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...

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ژورنال

عنوان ژورنال: Quarterly of Applied Mathematics

سال: 1944

ISSN: 0033-569X,1552-4485

DOI: 10.1090/qam/10672