A generalization of Floater–Hormann interpolants
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In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on γ, an additional positive integer parameter. For γ = 1, the original Floater–Hormann interpolants are obtained. When γ>1 we prove that the new rational functions share a lot of the nice properties of the original Floater–Hormann functions. Indeed, for any configuration of nodes, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum (h^*) and maximum (h) distance between two consecutive nodes. It turns out that, in contrast to the original Floater-Hormann interpolants, for all γ > 1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h∼ h^*). In such cases, we also estimate the uniform and the pointwise approximation errors for functions having different degree of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater-Hormann interpolants.
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