Polynomials with coefficients from a finite set
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- by Peter Borwein, Tamás Erdélyi and Friedrich Littmann PDF
- Trans. Amer. Math. Soc. 360 (2008), 5145-5154
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 $\mathbb {C}$ 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 $\mathbb {C}$ 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 $n$ be so large that \[ \delta _n:=33\pi \frac {\log n}{\sqrt {n}} \] satisfies $\delta _n\le 1$. We show that any polynomial in \begin{align*} K_n&:=\Big \{\sum _{k=0}^n a_k z^k : |a_0|=|a_n|=1\text { and }|a_k|\le 1\Big \} \end{align*} has at least \[ 8\sqrt {n}\log n \] zeros in any disk with center on the unit circle and radius $\delta _n$.
References
- Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- P. Borwein and T. Erdélyi, Littlewood-type problems on subarcs of the unit circle, Indiana Univ. Math. J. 46 (1997), no. 4, 1323–1346. MR 1631600, DOI 10.1512/iumj.1997.46.1435
- B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 865–889 (English, with English and French summaries). MR 1779897
- John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449, DOI 10.1007/978-1-4612-0817-4
- P. Dienes, The Taylor series: an introduction to the theory of functions of a complex variable, Dover Publications, Inc., New York, 1957. MR 0089895
- R. J. Duffin and A. C. Schaeffer, Power series with bounded coefficients, Amer. J. Math. 67 (1945), 141–154. MR 11322, DOI 10.2307/2371922
- Tamás Erdélyi, On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), no. 1, 97–111. MR 1827077, DOI 10.1307/mmj/1008719037
- P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105–119. MR 33372, DOI 10.2307/1969500
- Sergei Konyagin, On a question of Pichorides, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 385–388 (English, with English and French summaries). MR 1440953, DOI 10.1016/S0764-4442(97)80072-9
- John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0244463
- G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187, DOI 10.1142/1284
- A. M. Odlyzko and B. Poonen, Zeros of polynomials with $0,1$ coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317–348. MR 1252071
- Yuval Peres and Boris Solomyak, Approximation by polynomials with coefficients $\pm 1$, J. Number Theory 84 (2000), no. 2, 185–198. MR 1795789, DOI 10.1006/jnth.2000.2514
- Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
- H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
- G. Szegö, Tschebyscheffsche Polynome und nichtfortsetzbare Potenzreihen, Math. Ann. 87 (1922), no. 1-2, 90–111 (German). MR 1512103, DOI 10.1007/BF01458039
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
- Received by editor(s): June 8, 2005
- Received by editor(s) in revised form: February 15, 2006
- Published electronically: May 27, 2008
- © Copyright 2008 by the authors
- Journal: Trans. Amer. Math. Soc. 360 (2008), 5145-5154
- MSC (2000): Primary 30B30; Secondary 11C08, 30C15
- DOI: https://doi.org/10.1090/S0002-9947-08-04605-9
- MathSciNet review: 2415068