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On the minimum moduli of normalized polynomials

Approximation And Interpolation Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1105)

Abstract

Consider any complex polynomial pn(z)=1+\(\mathop \Sigma \limits_{j = l}^n\) ajzj which satisfies \(\mathop \Sigma \limits_{j = l}^n\)|aj|=1, and let Γn denote the supremum of the minimum moduli on |z|=1 of all such polynomials pn(z). We show that

$$1 - \frac{1}{n} \leqslant \Gamma _n \leqslant \sqrt {1 - \frac{1}{n}} , for all n \geqslant 1.$$

If the coefficients of pn(z) are further restricted to be positive numbers and if \(\tilde \Gamma _n\) denotes the analogous supremum of the minimum modulion |z|=1 of such polynomials, we similarly show that

$$1 - \frac{1}{n} \leqslant \tilde \Gamma _n \leqslant \sqrt {1 - \frac{3}{{\left( {2n + 1} \right)}}} , for all n \geqslant 1.$$

We also include some recent numerical experiments on the behavior of Γn, as well as some related conjectures.

Keywords

  • Richardson Extrapolation
  • Minimum Modulus
  • Multiple Precision
  • Related Conjecture
  • Multiplier Problem

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.

Research supported by the Air Force Office of Scientific Research, the Department of Energy, and the Alexander von Humboldt Stiftung.

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References

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© 1984 Springer-Verlag

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Ruscheweyh, S., Varga, R.S. (1984). On the minimum moduli of normalized polynomials. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072408

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  • DOI: https://doi.org/10.1007/BFb0072408

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

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