On the divergence of polynomial interpolation

Abstract

Consider a triangular interpolation scheme on a continuous piecewise C 1 curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ out the subset of Γ outside of the convergence region. In the paper we show that the sequence of interpolating polynomials, { P n } n , is divergent on all the points of Γ out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of { P n ( z)} n that converges to f( z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure. The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence { P n ( z)} n is divergent.