Localized approximation by rational interpolation at equidistant nodes

We consider the problem of interpolating a given function on arbitrary configurations of nodes in a compact interval, with a special focus on the case of equidistant or quasi-equidistant nodes. In this case, instead of polynomial interpolation, a family of rational interpolants introduced by Floater and Hormann in [2] turns out to be very useful . Such interpolants (briefly FH interpolants) generalize Berrut's rational interpolation [1] introducing a fixed integer parameter d >= 1 to speed up the convergence getting, in theory, arbitrarily high approximation orders. In this talk we will further generalize by presenting a whole new family of rational interpolants that depend on an additional parameter ? ? N. When ? = 1 we get the original FH interpolants. For ? > 1 we will see that the new interpolants share a lot of the interesting properties of the original FH interpolants (no real poles, baryentric-type representation, high rates of approximation). But, in addition, we get uniformly bounded Lebesgue constants and a more localized approximation of less smooth functions, compared to the original FH interpolation.