Hermite-Pade approximants for systems of Markov-type functions

Padé approximation has two natural extensions to vector rational approximation through the so called type I and type II Hermite-Padé approximants. The convergence properties of type II Hermite-Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Padé approximants for Nikishin systems of functions.

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“…The convergence for more general sequences of multi-indices was treated in [6], [7] and [12]. In [10, Lemma 2.9] it was shown that if (σ 1 , .…”

Section: Introductionmentioning

confidence: 99%

On the convergence of type I Hermite–Padé approximants

Lagomasino

Peralta

2015

Advances in Mathematics

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“…Since Ψ n,0 = Q n , and deg Q n ≤ |n| = N n,1 , we obtain (for details, see Proposition 3 in [9]) Lemma 3 For each k = 1, . .…”

Section: Applicationsmentioning

confidence: 99%

“…For the rest of the indicated values of k, the formula may be proved by induction for decreasing values of the index k (for details, see Proposition 2 in [9]). …”

Section: Applicationsmentioning

confidence: 99%

“…In [9] (see also [1]), the more general construction of mixed Angelesco-Nikishin systems (or generalized Nikishin systems) is studied and the n-th root asymptotic behavior of their Hermite-Padé approximants is given. We adopt the notation introduced in [9] and use some of the new orthogonality relations revealed in that paper. Nevertheless, we restrict our attention to purely Nikishin systems because our present methods do not cover the generalized case.…”

Section: Applicationsmentioning

confidence: 99%

“…. , n 2 − 1 (for details see [2], [4] and [9]). We study the asymptotic behavior of the sequences {Q n } and {w n } as well as of the varying measures with respect to which these polynomials are orthogonal when n 1 = n 2 → ∞.…”

Section: Introductionmentioning

confidence: 99%

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Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

Ysern

Lagomasino

1999

Constr. Approx.

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It is known that the common denominator of the Hermite-Padé approximants of a mixed Angelesco-Nikishin system shares orthogonality relations with respect to each function in the system. It is less known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures.

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Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

Aptekarev

Assche

Yattselev

2016

Comm Pure Appl Math

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Abstract. Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).

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Hermite-Pade approximants for systems of Markov-type functions

Cited by 134 publications

References 8 publications

On the convergence of type I Hermite–Padé approximants

On the convergence of type I Hermite–Padé approximants

Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

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