Rational Approximations and
                    Orthogonality

Никишин, Е. М.; Сорокин, В. Н.

doi:10.1090/mmono/092

Cited by 351 publications

(402 citation statements)

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“…An explanation of why this lemma follows from the results in [7] is contained in section 4 of [1]. The vector measure µ ∈ M 1 is called the equilibrium solution for the vector potential problem determined by the interaction matrix C on the system…”

Section: Uniqueness Is a Desirable Condition A Multi-indexmentioning

confidence: 99%

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Fourier–Padé Approximants for Nikishin Systems

Lagomasino

Ceniceros

2008

Constr Approx

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Abstract. We study type I Fourier-Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I HermitePadé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector valued equilibrium problems for the logarithmic potential.

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Section: Uniqueness Is a Desirable Condition A Multi-indexmentioning

confidence: 99%

“…, where The corresponding result for Hermite-Padé approximants of Nikishin systems appears in Section 7, Chapter 5 of [7] (see also [6]). …”

Section: Uniqueness Is a Desirable Condition A Multi-indexmentioning

confidence: 99%

Fourier–Padé Approximants for Nikishin Systems

Lagomasino

Ceniceros

2008

Constr Approx

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“…. , r): see [9,10,12] for detailed expositions of this well-known connection based upon the orthogonality property…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The expression of this II Padé approximants is well-known and goes back to Hermite [7], who used them for proving the transcendency of e. For example in [9], one finds the following integral expressions of Hermite:…”

Section: Indeed With Kmentioning

confidence: 99%

Remainder Pade Approximants for the Exponential Function

Prévost

Rivoal

2006

Constr Approx

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Abstract. Following earlier research of ours, we propose a new method for obtaining the complete Padé table of the exponential function. It is based on an explicit construction of certain Padé approximants not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and non trivial coincidence is proved more generally for type II simultaneous Padé approximants for a family (exp(a j z)) j=1,...,r with distinct complex a's and we recover Hermite's classical formulae. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesú, Coussement and van Assche, which generalise the classical Charlier orthogonal polynomials.

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“…There are many equivalent ways of defining the logarithmic capacity of a compact set, the most commonly used are through the transfinite diameter (see [10], § 5.5), the Tchebyshev constant ( [10], § 5.5), the Robin constant ( [10], § 5.1), and minimal energy ( [7], § 5.3). For definiteness, we give one definition and urge the reader to look in the references above if more details are needed.…”

Section: Types Of Convergence 221 Convergence In Capacitymentioning

confidence: 99%