Numerische Mathematik Manuscript-nr. Polynomial Interpolation of Minimal Degree

A note on interpolation, best approximation, and the saturation property

In this note, we prove that the well known saturation assumption implies that piecewise polynomial interpolation and best approximation in finite element spaces behave in similar fashion. That is, the error in one can be used to estimate the error in the other. We further show that interpolation error can be used as an a posteriori error estimate that is both reliable and efficient.

Numerical integration over a disc. A new Gaussian quadrature formula

The ordinary type of information data for approximation of functions f or functionals of them in the univariate case consists of function values {f(x1), . . . , f(xm)}. The classical Lagrange interpolation formula and the Gauss quadrature formula are famous examples. The simplicity of the approximation rules, their universality, the elegancy of the proofs and the beauty of these classical resul...

Error bounds for approximation in Chebyshev points

This paper improves error bounds for Gauss, Clenshaw-Curtis and Fejér’s first quadrature by using new error estimates for polynomial interpolation in Chebyshev points. We also derive convergence rates of Chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals. Preliminary numerical results show that the improved error bounds are ...

Error bounds for minimal energy bivariate polynomial splines

Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of th...