A study of lebesgue constants in barycentric rational and multivariate polynomial interpolation

Abstract

The Lebesgue constant is a valuable numerical instrument for linear interpolation, because it indicates how the interpolant of a function compares to the best linear approximant of that function. Furthermore, if the interpolant is computed by making use of the Lagrange basis functions, then the Lebesgue constant also expresses the conditioning of the interpolation problem at hand. Many publications have been devoted to the search for optimal interpolation points, optimal in the sense that these points lead to a minimal Lebesgue constant for interpolation problems on the interval [-1,1]. In this thesis, the best results obtained in univariate polynomial interpolation are generalized to univariate rational interpolation. In addition, this generalization provides a very practical and useful result in the case of barycentric rational interpolation, where simple equidistant interpolation points apparently yield very slowly increasing Lebesgue constants. The literature demonstrates a direct link between the orthogonality of polynomials and optimal interpolation points for polynomial interpolation. In this thesis, this connection is further explored for the case of linear interpolation, using rational functions with a predetermined denominator (preassigned poles) on the one hand and multivariate polynomial functions on the unit disk on the other hand.