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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Power series with restricted coefficients
and a root on a given ray


Authors: Franck Beaucoup, Peter Borwein, David W. Boyd and Christopher Pinner
Journal: Math. Comp. 67 (1998), 715-736
MSC (1991): Primary 30C15; Secondary 30B10, 12D10
MathSciNet review: 1468939
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Abstract: We consider bounds on the smallest possible root with a specified argument $\phi$ of a power series $f(z)=1+{ \sum _{n=1}^{\infty}} a_{i}z^{i}$ with coefficients $a_{i}$ in the interval $[-g,g]$. We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when $\phi/2\pi$ is rational. When $g\geq 2\sqrt{2}+3$ we show that the smallest disc containing two roots has radius $(\sqrt{g}+1)^{-1}$ coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller $g$. We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates, $\lambda$, of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of $[-1/2,1/2)$ power series; in particular we obtain the sharp inequality $|\lambda |\leq 3/2$.


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Additional Information

Franck Beaucoup
Affiliation: Equipe de Mathématiques appliquées, Ecole des Mines de Saint-Etienne, 42023 Saint-Etienne, France
Email: beaucoup@emse.fr

Peter Borwein
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
Email: pborwein@cecm.sfu.ca

David W. Boyd
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: boyd@math.ubc.ca

Christopher Pinner
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada & Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: pinner@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00960-0
PII: S 0025-5718(98)00960-0
Keywords: Power series, restricted coefficients, beta-numbers
Received by editor(s): July 15, 1996
Additional Notes: Research of the second and third authors was supported by the NSERC
Article copyright: © Copyright 1998 by the authors