Polynomials with coefficients from a finite set
Authors:
Peter Borwein, Tamás Erdélyi and Friedrich Littmann
Journal:
Trans. Amer. Math. Soc. 360 (2008), 51455154
MSC (2000):
Primary 30B30; Secondary 11C08, 30C15
Published electronically:
May 27, 2008
MathSciNet review:
2415068
Fulltext PDF Free Access
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Abstract: In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let be so large that satisfies . We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius .
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Additional Information
Peter Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email:
pborwein@cecm.sfu.ca
Tamás Erdélyi
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email:
terdelyi@math.tamu.edu
Friedrich Littmann
Affiliation:
Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105
Email:
Friedrich.Littmann@ndsu.edu
DOI:
http://dx.doi.org/10.1090/S0002994708046059
PII:
S 00029947(08)046059
Keywords:
Zeros,
rational functions,
DuffinSchaeffer Theorem,
Littlewood polynomials
Received by editor(s):
June 8, 2005
Received by editor(s) in revised form:
February 15, 2006
Published electronically:
May 27, 2008
Article copyright:
© Copyright 2008
by the authors
