A de montessus theorem for vector valued rational interpolants

Graves‐Morris, P. R.; Saff, E. B.

doi:10.1007/bfb0072414

Cited by 36 publications

(41 citation statements)

References 4 publications

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“…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.…”

Section: Introductionmentioning

confidence: 96%

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

Sidi

2008

Isr. J. Math.

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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.

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Section: Introductionmentioning

confidence: 96%

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

Sidi

2008

Isr. J. Math.

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“…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 .…”

Section: Hermite-padé Approximantsmentioning

confidence: 99%

Zero distribution of incomplete Padé and Hermite–Padé approximations

Ysern

Ceniceros

2016

Journal of Approximation Theory

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We prove that a subsequence of the normalized zero counting measures of incomplete Padé approximants tend, as the degree of the numerators goes to infinity and that of the denominators remains bounded, to the uniform distribution on the largest circle centered at the origin inside of which the approximants converge in the sense of the Hausdorff content. As a consequence, a Jentzsch-Szegő type theorem for row sequences of Hermite-Padé approximants is given.

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“…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.…”

Section: Introductionmentioning

confidence: 96%

Row Convergence Theorems for Vector-Valued Padé Approximants

Graves‐Morris

Iseghem²

1997

Journal of Approximation Theory

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Yet another method of proof of de Montessus' 1902 theorem is given. We show how this proof readily extends to row convergence theorems for four different kinds of vector Pade approximants. These approximants all belong to the category associated with vector-valued C-fractions formed using generalised inverses. The proof of a conjecture by Graves-Morris and Saff (J. Comput. Appl. Math. 23, 1988, 63 85) is given and new row convergence theorems for hybrid vector Pade approximants are proved.1997 Academic Press

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A de montessus theorem for vector valued rational interpolants

Cited by 36 publications

References 4 publications

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

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

Zero distribution of incomplete Padé and Hermite–Padé approximations

Row Convergence Theorems for Vector-Valued Padé Approximants

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