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Behavior of the lagrange interpolants in the roots of unity

Part of the Lecture Notes in Mathematics book series (LNM,volume 1435)

Abstract

Let A 0 be the class of functions f analytic in the open unit disk |z| < 1, continuous on |z| ≤ 1, but not analytic on |z| ≤ 1. We investigate the behavior of the Lagrange polynomial interpolants L n−1(f, z) to f in the n-th roots of unity. In contrast with the properties of the partial sums of the Maclaurin expansion, we show that for any w, with |w| > 1, there exists a gA 0 such that L n−1(g, w) = 0 for all n. We also analyze the size of the coefficients of L n−1(f, z) and the asymptotic behavior of the zeros of the L n−1(f, z).

Keywords

  • Unit Circle
  • Algebraic Polynomial
  • Open Unit Disk
  • Maclaurin Series
  • Good Approxi

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The research of this author was conducted while visiting the University of South Florida.

The research of this author was supported, in part, by the National Science Foundation under grant DMS-881-4026.

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Dedicated to R.S. Varga on the occasion of his sixtieth birthday.

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© 1990 Springer-Verlag

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Ivanov, K.G., Saff, E.B. (1990). Behavior of the lagrange interpolants in the roots of unity. In: Ruscheweyh, S., Saff, E.B., Salinas, L.C., Varga, R.S. (eds) Computational Methods and Function Theory. Lecture Notes in Mathematics, vol 1435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087899

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  • DOI: https://doi.org/10.1007/BFb0087899

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52768-8

  • Online ISBN: 978-3-540-47139-4

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