Generalizing Floater Hormann interpolation

As known, polynomial interpolation is not advisable in the case of equidistant nodes, given the exponential growth of the Lebesgue constants and the consequent stability problems. In [1] Floater and Hormann introduce a family of rational interpolants (briefly FH interpolants) depending on a fixed integer parameter d >= 1. They are based on any configuration of the nodes in [a, b], have no real poles and approximation order O(h^{d+1}) for functions in C^{d+2}[a, b], where h denotes the maximum distance between two consecutive nodes. FH interpolants turn out to be very useful for equidistant or quasi-equidistant configurations of nodes when the Lebesgue constants present only a logarithmic growth as the number of nodes increases [2, 3]. In this talk, we introduce a generalization of FH interpolants depending on an additional parameter ? ? N. If ? = 1 we get the classical FH interpolants, but taking ? > 1 we succeed in getting uniformly bounded Lebesgue constants for quasi-equidistant configurations of nodes. Moreover, in comparison with the original FH interpolants, we show that the new interpolants present a much better error prole when the function is less smooth.