Incomplete Padé approximation and convergence of row sequences of Hermite–Padé approximants

Abstract

We give a Montessus de Ballore type theorem for row sequences of Hermite-Padé approximations of vector valued analytic functions refining some results in this direction due to P.R. Graves-Morris and E.B. Saff. We do this introducing the notion of incomplete Padé approximation which contains, in particular, simultaneous Padé approximation and may be applied in the study of other systems of approximants as well.2010 Mathematics Subject Classification. Primary 41A21, 41A28; Secondary 41A25.

Zero distribution of incomplete Padé and Hermite–Padé approximations

“…was proved in [6], where A n,m is some constant and ϕ n,m−m * is a polynomial of degree less than or equal to m − m * normalized as in (6). The numbers A n,m determine the convergence in σ-content of the specific sequence of approximants {r n,m }.…”

“…The numbers A n,m determine the convergence in σ-content of the specific sequence of approximants {r n,m }. Actually (see [6,Theorem 3.4]), it holds that…”

“…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 .…”

“…Incomplete Padé approximants have been recently introduced to study direct and inverse results in simultaneous approximation, see [6,7]. They are defined in a similar manner as the Padé approximants except for the fact that the number of interpolating conditions may be fewer than the number of coefficients to be determined.…”

“…The number R * m (f ) can be defined in terms of the coefficients appearing in (1) and ii), see formulas (7) and (8) in Section 3 below. Notice the different starting point for the definition of R * m (f ) in [6]. It was also shown that R * m (f ) may take any value between R m * (f ) and R m (f ) depending on the specific sequence of incomplete Padé approximants considered so that a characterization of R * m (f ) in terms of intrinsic properties of the function f is not possible.…”