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

Abstract

In a recent paper of the author [8], 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. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F (z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.

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

“…The following lemma is the same as Lemma 3.4 in [12], with the exception of (3.6), which can be proved by invoking (3.2) and (3.5) in (D i,n , D m,n ). …”

“…The following theorem gives a closed-form expression for Q(z) in simple terms, and is the analogue of Theorem 3.6 in [12].…”

“…We now recall some technical tools that were used in [12], and will be used throughout this work as well. The following lemma was stated and proved as Lemma A.1 in [13].…”

“…The next lemma, which is Lemma 3.5 in [12], gives the determinant representation of the error function F(z) − R p,k (z), and we will be analyzing it in what follows.…”

“…In yet another recent work [12], we continued to study one of the three interpolation procedures that was denoted IMMPE in [10]. In particular, we studied the convergence properties of IMMPE as it is being applied to vectorvalued meromorphic functions F(z) that have simple poles.…”