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On Uniform Approximation of Rational Perturbations of Cauchy Integrals

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On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Content Provider	Springer Nature Link
Author	Yattselev, Maxim
Copyright Year	2010
Abstract	Let [c, d] be an interval on the real line and μ be a measure of the form $$d\mu = \dot \mu d\omega _{\left[ {c,d} \right]}$$ with $$\dot \mu = h\hbar$$ , where $$\hbar (t) = (t - c)^{\alpha _c } (d - t)^{\alpha _d }$$ , α $_{ c }$, α $_{ d }$ ∈ [0, 1/2), h is a Dini-continuous non-vanishing function on [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 $[c,d]\cap z(q)=\emptyset$ , where z(q) is the set of the zeros of q. We show that AAK-type meromorphic as well as diagonal multipoint Padé approximants to $$f(z):= \int{d\mu(t)\over z-t} + \left({p\over q}\right)(z)$$ converge locally uniformly to $$\mathfrak{f}$$ in $$\mathfrak{D}_\mathfrak{f} \cap \mathbb{D}$$ and $$\mathfrak{D}_\mathfrak{f}$$ , respectively, where $$\mathfrak{D}_\mathfrak{f}$$ is the domain of analyticity of $$\mathfrak{f}$$ and $$\mathbb{D}$$ 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 density $$\dot \mu$$ to vanish on (c, d), although in a strictly controlled manner.
Starting Page	1
Ending Page	33
Page Count	33
File Format	PDF
ISSN	16179447
Journal	Computational Methods and Function Theory
Volume Number	10
Issue Number	1
e-ISSN	21953724
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-09-09
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Approximation by rational functions Computational Mathematics and Numerical Analysis Strong asymptotics non-Hermitian orthogonality Functions of a Complex Variable Orthogonal functions and polynomials, general theory Approximation by other special function classes Analysis multipoint Padé approximation rational approximation meromorphic approximation Padé approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Theory and Mathematics Analysis

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On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities

Fourier–Padé Approximants for Nikishin Systems

Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

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On Uniform Approximation of Rational Perturbations of Cauchy Integrals

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