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On approximation and interpolation of entire functions with index-pair \((p,q)\). (English) Zbl 0831.30023

Let \(E\) be a compact set in the complex plane, and let \(f\) be analytic on \(E\) with \(|f |= \sup_{z \in E} |f(z) |\). Let \(L_n\) be the Lagrange interpolation polynomial interplating \(f\) in \((n + 1)\)th Fekete points of \(E\). The authors investigate the behaviour of the “approximation errors” \(\mu_{n,1} (f) = \inf_p |f - p |\), where \(p\) is a polynomial of degree at most \(n\), \(\mu_{n,2} (f) = |L_n - f |\), \(\mu_{n,3} (f) = |L_n - L_{n - 1} |\) and they use the concept of order and type of entire functions introduced by O. P. Juneja, G. P. Kapoor and S. K. Bajpai [J. Reine Angew. Math. 282, 53-67 (1976; Zbl 0321.30031)] to prove some relations between \(\mu_{n,j}\) and these orders and types, e.g.: If \(f\) can be extended to an entire function of certain order and type (in the above sense), then \(\sum^\infty_{n = 0} \mu_{n,j} z^{n + 1} (j = 1,2,3)\) is an entire function of same order and type.
Reviewer: K.Menke (Dortmund)

MSC:

30E10 Approximation in the complex plane
30E15 Asymptotic representations in the complex plane

Citations:

Zbl 0321.30031
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