A de montessus theorem for vector valued rational interpolants

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

“…Our theory is in the spirit of that given by Saff [4] for the scalar rational interpolation problem and by Graves-Morris and Saff [2] for vector-valued rational interpolants, while the technique used here is analogous to that developed by Sidi, Ford, and Smith [10] and used by Sidi [5] in the study of Padé approximants, hence different from that of [4] and [2] . In addition, the technique we use here enables us to obtain optimally refined results in the form of asymptotic equalities.…”

Zero distribution of incomplete Padé and Hermite–Padé approximations

“…Diagonal sequences (when m 1 = · · · = m d = m, n = (d + 1) m, m ∈ N) have been profusely studied as well as their applications in different fields (number theory, random matrices, Brownian notions, Toda lattices, etc...). Regarding row sequences, the seminal paper is [13], where an analogue of the Montessus de Ballore theorem [17] is proved. Further developments may be found in [6,7], where the incomplete Padé approximants are introduced and applied to the characterization of the convergence of Q n,m with geometric rate in terms of intrinsic properties of the system of functions f .…”

Row Convergence Theorems for Vector-Valued Padé Approximants

“…One category is alternatively called simultaneous Pade approximants [1,2]. Analogues of de Montessus' theorem for these approximants have been established [11,24]. In this paper, we are concerned with the other category, in which vector inverses are directly or indirectly involved.…”