Abstract. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.

We give a Montessus de Ballore type theorem for row sequences of Hermite-Padé approximations of vector valued analytic functions refining some results in this direction due to P.R. Graves-Morris and E.B. Saff. We do this introducing the notion of incomplete Padé approximation which contains, in particular, simultaneous Padé approximation and may be applied in the study of other systems of approximants as well.2010 Mathematics Subject Classification. Primary 41A21, 41A28; Secondary 41A25.

It is known that the common denominator of the Hermite-Padé approximants of a mixed Angelesco-Nikishin system shares orthogonality relations with respect to each function in the system. It is less known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures.

The strong asymptotic behaviour of orthogonal polynomials with respect to a general class of varying measures is given for the case of the unit circle and the real line. These results are used to obtain certain asymptotic relations for the polynomials involved in the construction of Hermite-Pade approximants of a Nikishin system of functions.

Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. Mathematics Subject Classification (2000)65D32 · 41A20 · 41A80