Best Meromorphic Approximation of Markov Functions on the Unit Circle

Baratchart, Laurent; Prokhorov, Vasiliy A.; Saff, Edward B.

doi:10.1007/s002080010015

Cited by 10 publications

(7 citation statements)

References 37 publications

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“…It is an immediate consequence of the corresponding properties of D ρ (Y ; •) and ϕ that D y is outer, has non-tangential continuous boundary values on both sides of [c, d] and [c, d] * whenever y is a Dinicontinuous pair 4 , has winding number zero on any curve in D ∩ D * , and satisfies…”

Section: Proofs Of Theorems 1-4mentioning

confidence: 96%

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On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Yattselev

2009

Comput. Methods Funct. Theory

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Let [c, d] be an interval on the real line and µ be a measure of the form dµ =μdω [c,d] with an argument of bounded variation, and ω [c,d] is the normalized arcsine distribution on [c, d]. Further, let p and q be two polynomials such that deg(p) < deg(q) and [c, d] ∩ z(q) = ∅, where z(q) is the set of the zeros of q. We show that AAK-type meromorphic as well as diagonal multipoint Padé approximants toconverge locally uniformly to f in D f ∩ D and D f , respectively, where D f is the domain of analyticity of f and D is the unit disk. In the case of Padé approximants we need to assume that the interpolation scheme is "nearly" conjugate-symmetric. A noteworthy feature of this case is that we also allow the densityμ to vanish on (c, d), even though in a strictly controlled manner.Mathematics Subject Classification (2000). 42C05, 41A20, 41A21, 41A30.

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Section: Proofs Of Theorems 1-4mentioning

confidence: 96%

“…q ≡ 1 and necessarily p ≡ 0, (such functions f are called Markov functions). The general case p ∈ [1, ∞] was addressed in [4], where again only Markov functions were considered and uniform convergence was shown under the assumption that µ belongs to the Szegő class, i.e. log(dµ(t)/dt) is integrable on [c, d].…”

Section: Introductionmentioning

confidence: 99%

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Yattselev

2009

Comput. Methods Funct. Theory

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“…Again, this turns out to be a rather typical problem of approximation theory, at least when restricting the poles of q n to belong to some Jordan curve surrounding [−1, 1]. A natural choice being an ellipse with foci at ±1, see also [45,6].…”

Section: Remarkmentioning

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 [16,45,6].…”

Section: Introductionmentioning

confidence: 99%

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

Mattei¹,

Milton²,

Putinar³

2020

Preprint

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In two phase materials, each phase having a non-local response in time, it has been found that for appropriate 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 find how the appropriate driving fields may be designed, we obtain 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.

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“…Best meromorphic approximants to Markov functions were studied per se by E. B. Saff, V. Prokhorov and the first author in [8]. Using results from [4] to make connection with orthogonality, these authors prove (and give error rates for) the uniform convergence of such approximants, locally uniformly on C \ I, whenever 1 ≤ p ≤ ∞ provided that λ satisfies the Szegő condition: log dλ/dt ∈ L 1 (I).…”

Section: Introductionmentioning

confidence: 99%

Meromorphic approximants to complex Cauchy transforms with polar singularities

Baratchart¹,

Yattselev²

2009

Sb. Math.

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We study AAK-type meromorphic approximants to functions of the formwhere R is a rational function and λ is a complex measure with compact regular support included in (−1, 1), whose argument has bounded variation on the support. The approximation is understood in L p -norm of the unit circle, p ≥ 2. We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of λ relative to the unit disk, that the approximants themselves converge in capacity to F , and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.Mathematics Subject Classification (2000). primary 41A20, 41A30, 42C05; secondary 30D50, 30D55, 30E10, 31A15.

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Best Meromorphic Approximation of Markov Functions on the Unit Circle

Cited by 10 publications

References 37 publications

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

Meromorphic approximants to complex Cauchy transforms with polar singularities

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