Browsing by Subject "Non-Hermitian orthogonality"

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    Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight
    (Springer, 2020) Barhoumi, Ahmad; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    We 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.
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    NUTTALL’S THEOREM WITH ANALYTIC WEIGHTS ON ALGEBRAIC S-CONTOURS
    (Elsevier, 2015-02) Yattselev, Maxim L.; Department of Mathematical Sciences, School of Science
    Given a function f holomorphic at infinity, the nth diagonal Padé approximant to f, denoted by [n/n]f, is a rational function of type (n,n) that has the highest order of contact with f at infinity. Nuttall’s theorem provides an asymptotic formula for the error of approximation f−[n/n]f in the case where f is the Cauchy integral of a smooth density with respect to the arcsine distribution on [−1,1]. In this note, Nuttall’s theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).
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    Strong Asymptotics of Jacobi-Type Kissing Polynomials
    (Taylor & Francis, 2021) Barhoumi, Ahmad; Mathematical Sciences, School of Science
    We investigate asymptotic behaviour of polynomials pnω(z) satisfying varying non-Hermitian orthogonality relations ∫−11xkpnω(x)h(x)eiωxdx=0,k∈0,…,n−1, where h(x)=h∗(x)(1−x)α(1+x)β, ω=λn, λ≥ 0 and h(x) is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials ( α=β=0) introduced in Asheim et al. [A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Preprint, 2012 Dec 6. arXiv:1212.1293] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in Celsus and Silva [Supercritical regime for the kissing polynomials. J Approx Theory. 2020 Mar 18;225:Article ID: 105408]; Deaño [Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. J Approx Theory. 2014 Oct 1;186:33–63], and depends heavily on those works.