ISSN:
1432-0940
Keywords:
Key words. Orthogonal polynomials, Asymptotic behavior, Padé and Padé-type approximants, Markov functions. AMS Classification. 42C05, 41A21, 30E10. 〈lsiheader〉 〈onlinepub〉8 May, 1998 〈editor〉Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;R.A. DeVore, E.B. Saff&lsilt;/a&lsigt; 〈pdfname〉14n2p209.pdf 〈pdfexist〉yes 〈htmlexist〉no 〈htmlfexist〉no 〈texexist〉yes 〈sectionname〉 〈/lsiheader〉
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Let p n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , \liminf_{n\to\infty}|p_n(z)|^{1/n}\ge e^{g_\Omega(z)} and, for a large class of measures μ , \lim_{n\to\infty}|p_n(z)|^{1/n}=e^{g_\Omega(z)}, where g Ω (z) is Green's function of $\Omega=\overline{{\bf C}}\backslash \Delta$ with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function f(z)=\int_{-1}^1{d\mu(t)\over z-t} to f on Ω with a certain geometric speed measured by g Ω (z) . We prove corresponding results when we restrict the freedom of p n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s003659900071
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