On rational approximation of Markov functions on finite sets

Prokhorov, Vasiliy A.

doi:10.1016/j.jat.2014.10.006

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…Again, this turns out to be a rather typical problem of approximation theory, at least when restricting the poles of to belong to some Jordan curve surrounding . A natural choice is an ellipse with foci at ; see also [6,53].…”

Section: Resultsmentioning

confidence: 99%

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An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

Mattei

Milton

Putinar

2022

Comm Pure Appl Math

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In two‐phase materials, each phase having a non‐local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [‐1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [‐1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [‐1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov‐type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time‐dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.

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Section: Resultsmentioning

confidence: 99%

“…In the same vein, the asymptotics of the optimal bound of our minimization problem inherently involves potential theory or operator theory concepts. We cite for a comparison basis a few remarkable results of the same flavor [6,22,53].…”

Section: ) Supmentioning

confidence: 99%

An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

Mattei

Milton

Putinar

2022

Comm Pure Appl Math

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On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

Ровба

Potseiko

2020

Math Notes

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On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev-Markov System

Ровба¹,

Rovba

Potseiko³

2020

Matematicheskie Zametki

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Рассматриваются приближения на отрезке $[-1,1]$ функций, представляющих собой комбинацию классических функций Маркова, частичными суммами рядов Фурье по системе рациональных дробей Чебышева-Маркова. Устанавливаются поточечная и равномерная оценки приближений. В случае, когда производная меры слабо эквивалентна некоторой степенной функции, установлены асимптотическое выражение мажоранты равномерных приближений и оптимальное значение параметра, обеспечивающее наибольшую скорость приближений используемым методом. В случае четной кратности полюсов аппроксимирующей функции асимптотическая оценка точна. Приводятся примеры аппроксимаций индивидуальных функций. Библиография: 17 названий.

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On rational approximation of Markov functions on finite sets

Cited by 3 publications

References 27 publications

An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev-Markov System

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