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An interpolation problem on the circle between Lagrange and Hermite problems. (English) Zbl 1357.41013

Polynomial and Laurent polynomial interpolation are studied in this paper, the task stated as a problem of interpolation on the circle. The provided data contain function values at all the given even number of points and derivative data on top of this at every other node, i.e., the ones with odd indices. The interpolation problem is considered under various aspects, that is, existence, numerical computation, convergence to continuous approximants including convergence rates, examples. The solutions of the Hermite type interpolation problems are provided in various fashions, namely the approximants are Laurent polynomials expressed as explicit linear combinations of barycentric expressions suitable to the given problem on the circle, and also as linear combinations of an orthogonal basis of the said space of Laurent polynomials.

MSC:

41A25 Rate of convergence, degree of approximation
41A05 Interpolation in approximation theory
65D05 Numerical interpolation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

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