1,720,978 research outputs found
On LR2-best rational approximants to Markov functions on several intervals
No full textLet f(z)=∫(z−x)−1dμ(x), where μ is a Borel measure supported on several subintervals of (−1,1) with smooth Radon–Nikodym derivative. We study strong asymptotic behavior of the error of approximation (f−rn)(z), where rn(z) is the LR2-best rational approximant to f(z) on the unit circle with n poles inside the unit disk
Non-Hermitian Orthogonality and Meromorphic Approximation
No full textIn this thesis we consider the distribution of poles of meromorphic and multipoint Padé approximants to the sum of a rational function and a Cauchy transform of a complex measure
Strong Asymptotics of Hermite-Padé Approximants for Angelesco Systems
No full textIn this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices
On an identity by Ercolani, Lega, and Tippings
Get PDFIn this note we prove that where and
are positive integers, which resolves a question posed by Ercolani, Lega, and
Tippings
Uniformity of Strong Asymptotics in Angelesco Systems
No full textLet \mu_1 and \mu_2 be two complex-valued Borel measures on the real line such that \operatorname{supp} \mu_1 =[\alpha_1,\beta_1] < \operatorname{supp} \mu_2 =[\alpha_2,\beta_2] and {\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}, where \rho_i(x) is the restriction to [\alpha_i,\beta_i] of a function non-vanishing and holomorphic in some neighborhood of [\alpha_i,\beta_i]. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices (n_1,n_2) tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to \min\{n_1,n_2\}
Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight
No full textWe investigate asymptotic behavior of polynomials Qn(z) satisfying non-Hermitian orthogonality relations
∫ΔskQn(s)ρ(s)ds=0,k∈{0,…,n−1},
where Δ:=[−a,a]∪[−ib,ib], a,b>0, and ρ(s) is a Jacobi-type weight
An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC
No full textLet {φi}∞i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say En(μ), of random polynomials
Pn(z):=∑i=0nηiφi(z),
where η0,…,ηn are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that En(|dξ|) admits an asymptotic expansion of the form
En(|dξ|)∼2πlog(n+1)+∑p=0∞Ap(n+1)−p
(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case En(μ) admits an analogous expansion with coefficients the Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same)
An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials
Get PDFLet {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say n(α), of random polynomials Pn(z):= ∑i=0nηiφi(z;α), where η0,…,ηn are i.i.d. standard Gaussian random variables. When α=0, φi(z;0)=zi and Pn(z) are called Kac polynomials. In this case it was shown by Wilkins that n(0) admits an asymptotic expansion of the form n(0)∼2πlog(n+1)+ ∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of (α) for α≠0. As it turns out, the leading term of the asymptotics in this case is (1∕π)log(n+1)
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