Direct and Inverse Results for Multipoint Hermite-Pade Approximants

Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Pade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1606.07920, arXiv:1801.03004, arXiv:1203.494

Computation of conformal representations of compact Riemann surfaces

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so called Nikishin systems of two measures.Comment: To appear in Mathematics of Computatio

Direct and Inverse Results on Row Sequences of Hermite-Padé Approximation

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.The work of B. de la Calle Ysern received support from MINCINN under grant MTM2009-14668-C02-02 and from UPM through Research Group “Constructive Approximation Theory and Applications”. The work of J. Cacoq and G. López was supported by Ministerio de Economía y Competitividad under grants MTM2009-12740-C03-01 and MTM2012-36372-C03-01

The Jacobi matrices approach to Nevanlinna-Pick problems

A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of \bR_0-functions gives rise to a linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show that $J$ is a positive operator. Then it is proved that the corresponding Nevanlinna-Pick problem has a unique solution iff the densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Pad\'e approximants to a unique solution $\varphi$ of the Nevanlinna-Pick problem converge to $\varphi$ locally uniformly in \dC\setminus\dR. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Pad\'e approximation for \bR_0-functions.Comment: 24 pages; Section 5 is modifed; some typos are correcte

Rational Approximations of Meromorphic Stieltjes-Type Functions

Available from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

Fourier-Padé Approximants for Nikishin systems

We study type I Fourier-Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite-Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential. © 2008 Springer Science+Business Media, LLC

On Perfect Nikishin Systems

We prove perfectness for Nikishin systems made up of three functions and apply this to the convergence of the associated Hermite-Pade approximants