Padé-Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $ S$-property of stationary compact sets

Gonchar, A A; Rakhmanov, Evguenii Andreevich; Suetin, Sergey Pavlovich

doi:10.1070/rm2011v066n06abeh004769

Cited by 25 publications

(42 citation statements)

References 51 publications

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“…The multipoint Padé approximant of order n of the function f (at the points in E n ) is the rational function R n = P n /Q n determined by (11) and by…”

Section: Statement Of the Problem And Discussionmentioning

confidence: 99%

“…To conclude this section we remark that in view of Theorem 1 the µ-minimizing compactum from the set K E,f with respect to the measure µ possesses an S-property. The existence of a compactum with the S-property is crucial not only in the study of distribution of zeros of orthogonal polynomials, but also in the derivation of the formulas of strong asymptotics, on the basis of the matrix Riemann-Hilbert method, and other methods as well (see [24], [34], [4], [1], [11], [9], [2], [23], [27], [14]).…”

Section: Statement Of the Problem And Discussionmentioning

confidence: 99%

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On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

Buslaev

Suetin

2016

Journal of Approximation Theory

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АннотацияFor an interval E = [a, b] on the real line, let µ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let V µ denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V −µ .Bibliography: 35 items.rational approximants, Padé approximation, orthogonal polynomials, distribution of poles, convergence in capacity Dedicated to the memory of Andrei Aleksandrovich Gonchar and Herbert Stahl NotationsThroughout the paper, we use the following notations.

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“…The multipoint Padé approximant of order n of the function f (at the points in E n ) is the rational function R n = P n /Q n determined by (11) and by…”

Section: Statement Of the Problem And Discussionmentioning

confidence: 99%

Section: Statement Of the Problem And Discussionmentioning

confidence: 99%

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

Buslaev

Suetin

2016

Journal of Approximation Theory

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“…Padé-Chebyshev approximation of a function f represented by series (1.1) is called (see, e.g., [3,16,30]) a rational fraction…”

Section: Introductionmentioning

confidence: 99%

“…Within such approach, it is natural to use the construction of Padé-Chebyshev approximations. Despite the problem relayed to the existence of a rational function π ch n, m (x; f ) (see [16]), the software Maple [11] allows one to realize namely nonlinear Padé-Chebyshev approximations. Such approximations are very frequently used in applications (see [12,13] …”

Section: Introductionmentioning

confidence: 99%

Padé approximants of special functions

2012

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Let fγ(x) = ∞ k=0 T k (x)/(γ) k , where (γ) k = γ(γ+1) · · · (γ+k−1) and T k (x) = cos (k arccos x) are Padé-Chebyshev polynomials. For such functions and their Padé-Chebyshev approximations π ch n,m (x; fγ), we find the asymptotics of decreasing the difference fγ(x) − π ch n, m (x; fγ) in the case where 0 m m(n), m(n) = o (n), as n → ∞ for all x ∈ [−1, 1]. Particularly, we determine that, under the same assumption, the Padé-Chebyshev approximations converge to fγ uniformly on the segment [−1, 1] with the asymptotically best rate.

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“…Presently, these techniques have been perfected in detail by Gonchar (see [2] and [3]) and Gonchar together with his students (see [4], [5], [6], [7], [8], [9], and [10]) in connection with problems of rational approximation with free poles. Rakhmanov [11] developed this approach for weak asymptotics of polynomials orthogonal with respect to discrete measures by suggesting describing the limit measure of the distribution of zeros of such discrete orthogonal polynomials in the form of the solution of the minimization problem for the energy of the logarithmic potential in the class of measures bounded by the counting measure of the support of the orthogonality measure.…”

Section: − Q =: ω(γ Q)mentioning

confidence: 99%

Asymptotics of Meixner polynomials and Christoffel–Darboux kernels

Aptekarev

Tulyakov

2013

Trans. Moscow Math. Soc.

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Abstract. We obtain the asymptotics of the classical Meixner polynomials (orthogonal with respect to a discrete measure supported at the nonnegative integer points) and the corresponding reproducing kernels (Christoffel-Darboux kernels) as the number n of the polynomial and the variable x tend to infinity under various relationships between their growth rates. (These asymptotics are known as the Plancherel-Rotach asymptotics.)

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Padé-Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $ S$-property of stationary compact sets

Cited by 25 publications

References 51 publications

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

Padé approximants of special functions

Asymptotics of Meixner polynomials and Christoffel–Darboux kernels

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