Local behaviour of polynomials
HTML articles powered by AMS MathViewer
- by D. P. Dryanov, M. A. Qazi and Q. I. Rahman PDF
- Math. Comp. 73 (2004), 1345-1364 Request permission
Abstract:
In this paper we study the local behaviour of a trigonometric polynomial $t(\theta ) := \sum _{\nu =-n}^{n} a_{\nu } e^{\mathrm {i}\nu \theta }$ around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in $[0 , 2 \pi )$. The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form $\sum _{\nu =0}^{n} a_{\nu } x^{\nu }$ are also proved.References
- N. I. Achieser, Theory of approximation, Frederick Ungar Publishing Co., New York, 1956. Translated by Charles J. Hyman. MR 0095369
- S.N. Bernstein, Sur une propriété des polynômes, Comm. Soc. Math. Kharkow Sér. 2 14 (1913), 1–6.
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- R. P. Boas Jr., Inequalities for polynomials with a prescribed zero, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 42–47. MR 0150269
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0465631
- M. Riesz, Formule d’interpolation pour la dérivée d’un polynôme, C. R. Acad. Sci. Paris 158 (1914), 1152–1154.
- M. Riesz, Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome, Jber. Deutsch. Math. Verein. 23 (1914), 354–368.
- I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z. 4 (1919), 271–287.
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
Additional Information
- D. P. Dryanov
- Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada
- Email: dryanovd@dms.umontreal.ca
- M. A. Qazi
- Affiliation: Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088
- Email: maqazi@tusk.edu
- Q. I. Rahman
- Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada
- Email: rahmanqi@dms.umontreal.ca
- Received by editor(s): August 20, 2002
- Received by editor(s) in revised form: December 22, 2002
- Published electronically: July 28, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 1345-1364
- MSC (2000): Primary 42A05, 26D05, 26D10, 30C10, 30A10
- DOI: https://doi.org/10.1090/S0025-5718-03-01585-0
- MathSciNet review: 2047090