A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure

Abstract

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177-187], three new interpolation procedures for vector-valued functions F(z), where F : C → C N , were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately.

A de Montessus type convergence study for a vector-valued rational interpolation procedure

“…, p. However, the quality of R p,k (z) as an approximation to F (z) in the z-plane depends heavily on how the c j are chosen. Thus, the methods IMPE, IMMPE, and ITEA choose the c j in special ways; as we have shown in [6], [7], and [8], the methods IMPE and IMMPE do provide very good approximations for meromorphic functions F (z). Here we prove that ITEA does too.…”

“…Some of the algebraic properties of these methods were already presented in [4] while others were explored in [5], where it was also shown that the methods are symmetric functions of the points of interpolation and that they reproduce vectorvalued rational functions exactly. In [6], [7], and [8], de Montessus and König type convergence theories for IMMPE and IMPE, as these methods are applied to vectorvalued meromorphic functions with simple poles, were presented. In this work, we treat the convergence properties of ITEA, as it is being applied to the same class of functions, and we prove de Montessus and König type theorems analogous to those for IMPE and IMMPE.…”

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

“…Using the Vieta formulas connecting the coefficients of a polynomial and its zeros it follows that there exists C 1 ≥ 1 such that (10) sup 10) and (11) imply that…”

“…In this second direction a significant contribution is due to Graves-Morris/Saff in [8] where they prove an analogue of the Montessus de Ballore theorem which plays a central role in the classical theory of Padé approximation. See also [9]- [10] for different approaches to the same type of results as well as [11] and references therein for least-squares versions.…”

Convergence of row sequences of simultaneous Fourier-Padé approximation

“…For example, we may have apriori knowledge of the location of some of the poles of f and we can use this information to fix some of the zeros of Q n,m at such points. Another possibility is to combine two (or more) approximation criteria to define the rational functions; for example, interpolation at a fixed number of points and the L 2 construction (or viceversa as in [8] and te references therein).…”