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Nuttall’s theorem with analytic weights on algebraic S-contours. (English) Zbl 1306.41005

Let \(f\) be a complex function holomorphic at \(\infty\). A diagonal Padé approximant of \(f\) is a rational function \(P_n/Q_n\), where \(P_n\), \(Q_n\) are polynomials of degree \(\leq n\) with \(Q_n(z)\,f(z) - P_n(z) = {\mathcal O}(z^{-n-1})\) as \(z \to \infty\). J. Nuttall [Constr. Approx. 6, No. 2, 157–166 (1990; Zbl 0685.41014)] has presented an asymptotic formula for the error \(f- P_n/Q_n\) in the case, where \(f\) is a Cauchy integral of a smooth density with respect to the arcsine distribution on \([-1,\,1]\). In this paper, the author extends Nuttall’s theorem to the case, where \(f\) is a Cauchy integral of analytic density on an algebraic S-contour.

MSC:

41A21 Padé approximation
30E15 Asymptotic representations in the complex plane
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane

Citations:

Zbl 0685.41014
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References:

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