13 research outputs found
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
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
General Results on the Convergence of Multipoint Hermite-Pade Approximants of Nikishin Systems
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
Hermite-Padé Approximation and Simultaneous Quadrature Formulas
We study the construction of a quadrature rule which allows the simultaneous integration of a given function with respect to different weights. This construction is built on the basis of simultaneous Padé approximation of a Nikishin system of functions. The properties of these approximants are used in the proof of convergence of the quadratures and positivity of the corresponding quadrature coefficients