Orthogonal polynomials with complex-valued weight function,I |
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Authors: | Herbert Stahl |
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Institution: | 1. TU-Berlin/Sekr. FR 6-8, Franklinstr. 28/29, 1000, Berlin 10, Fed. Rep. Germany
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Abstract: | In this paper we investigate the asymptotic behavior of polynomialsQ mn(z), m, n ∈ N, of degree ≤n that satisfy the orthogonal relation $$\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$ where/tf(z) is a function, which is supposed to be analytic on a continuum \(V \subseteq \hat C\) and all its singularities are supposed to be contained in a set \(E \subseteq \hat C\) of capacity zero, ω m+n (z) is a polynomial of degreem+n+1 with all its zeros contained inV, andC is a curve separatingV from the setE. We show that if the zeros of ω m+n have a certain asymptotic distribution form+n → ∞ and ifm/n ar 1, then the zeros of the polynomialsQ mn have a unique asymptotic distribution, which is closely related with the extremal domainD for single-valued analytic continuation of the functionf(z). The results are essential for the investigation of Padé and best rational approximants to the functionf(z). |
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