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Computational Methods and Function Theory 10 (2010), No. 1, 1--33
Copyright Heldermann Verlag 2010

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Maxim Yattselev
myattsel@sophia.inria.fr , INRIA, Project APICS, 2004 route des Lucioles --- BP 93, 06902 Sophia-Antipolis, France.

[Abstract-pdf] [Abstract-ps]

Let [c,d] be an interval on the real line and μ be a measure of the form dμ = \dot(&mu) dω[c,d] with \dot(&mu)=h\hbar, where \hbar(t)=(t-c)αc(d-t)αd, αc,αd∈[0,1/2), h is a Dini-continuous non-vanishing function on [c,d] with an argument of bounded variation, and ω[c,d] is the normalized arcsine distribution on [c,d]. Further, let p and q be two polynomials such that deg(p) < deg(q) and [c,d]∩zeros(q) = ∅, where zeros(q) is the set of the zeros of q. We show that AAK-type meromorphic as well as diagonal multipoint Padé approximants to
f(z) := ∫ dμ(t)/(z-t) + (p/q)(z)
converge locally uniformly to f in Df∩D and Df, respectively, where Df is the domain of analyticity of f and D is the unit disk. In the case of Padé approximants we need to assume that the interpolation scheme is "nearly" conjugate-symmetric. A noteworthy feature of this case is that we also allow the density \dot(&mu) to vanish on (c,d), although in a strictly controlled manner.

Keywords: Strong asymptotics, non-Hermitian orthogonality, meromorphic approximation, rational approximation, multipoint Padé approximation.

MSC 2000: 42C05, 41A20, 41A21, 41A30.

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