Formulas for bivariate hyperosculatory interpolation

Optimum-Point Formulas for Osculatory and Hyperosculatory Interpolation

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for f(x), over ( —1, 1), in terms of fixi), f'(xi) and f"(x/) at the irregularly-spaced Chebyshev points x¡ = —cos {(2i — l)ir/2n}, i = 1, • • ■ , n. The advantage over corresponding formulas for Xi equally spaced is in the squaring and cubing, in the respective osculatory and hyperoscul...

Error bounds for bivariate piecewise hermite interpolation

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...

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...

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. §