Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with nonpositive measures

Abstract

In this paper we prove the existence of real- and complex-valued measuresμ on the interval [−1,1] with the property that the diagonal Pade approximants [n/n],n=1,2,..., to the functionf(z)=∫dμ(x)/(x−z) neither converge at any fixed pointz∈C∼[−1,1] nor converge in capacity on any open (nonempty) setS inC∼[−1,1]. This result is derived from a theorem on the asymptotic behavior of orthogonal polynomials. It will be shown that it is possible to construct measuresμ. on [−1,1] such that for any arbitrarily prescribed asymptotic behavior there exist subsequences of the associated orthogonal polynomialsQ n that have this behavior.