Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients
In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random vari...
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| Published in: | Journal of mathematical analysis and applications Vol. 262; no. 2; pp. 554 - 563 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Elsevier Inc
15.10.2001
Elsevier |
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| ISSN: | 0022-247X, 1096-0813 |
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| Abstract | In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known. |
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| AbstractList | In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known. |
| Author | Farahmand, K. Grigorash, A. |
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| Keywords | Adler's theorem real roots random trigonometric polynomials coordinate transform complex roots random algebraic polynomials Jacobian of transformation density of zeros Number of complex zeros Zero of function Random polynomial Coodinate transformation Adler theorem Trigonometric polynomial Special function Jacobi matrix Algebraic polynomial Complex number |
| Language | English |
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| References | Ibragimov, Zeitouni (RF7) 1997; 349 Dunnage (RF4) 1968; 18 Anderson (RF2) 1984 Adler (RF1) 1981 Farahmand (RF5) 1997; 210 Farahmand (RF6) 1998 Sambandham (RF8) 1979; 81 Shepp, Vanderbei (RF9) 1995; 347 Wilkins (RF10) 1988; 103 Bharucha-Reid, Sambandham (RF3) 1986 Farahmand (10.1006/jmaa.2001.7554_RF5) 1997; 210 Adler (10.1006/jmaa.2001.7554_RF1) 1981 Ibragimov (10.1006/jmaa.2001.7554_RF7) 1997; 349 Bharucha-Reid (10.1006/jmaa.2001.7554_RF3) 1986 Wilkins (10.1006/jmaa.2001.7554_RF10) 1988; 103 Dunnage (10.1006/jmaa.2001.7554_RF4) 1968; 18 Sambandham (10.1006/jmaa.2001.7554_RF8) 1979; 81 Shepp (10.1006/jmaa.2001.7554_RF9) 1995; 347 Farahmand (10.1006/jmaa.2001.7554_RF6) 1998 Anderson (10.1006/jmaa.2001.7554_RF2) 1984 |
| References_xml | – year: 1998 ident: RF6 publication-title: Topics in Random Polynomials – year: 1981 ident: RF1 publication-title: The Geometry of Random Fields – year: 1986 ident: RF3 publication-title: Random Polynomials – volume: 81 start-page: 207 year: 1979 end-page: 215 ident: RF8 article-title: On the average number of real zeros of a class of random algebraic curves publication-title: Pacific J. Math. – volume: 210 start-page: 724 year: 1997 end-page: 730 ident: RF5 article-title: Complex roots of a random algebraic polynomial publication-title: J. Math. Anal. Appl. – volume: 103 start-page: 1249 year: 1988 end-page: 1258 ident: RF10 article-title: An asymptotic expansion for the expected number of real zeros of a random polynomial publication-title: Proc. Amer. Math. Soc. – volume: 347 start-page: 4365 year: 1995 end-page: 4383 ident: RF9 article-title: The complex zeros of random polynomials publication-title: Trans. Amer. Math. Soc. – volume: 18 start-page: 439 year: 1968 end-page: 460 ident: RF4 article-title: The number of real zeros of a class of random algebraic polynomials publication-title: Proc. London Math. Soc. – volume: 349 start-page: 2427 year: 1997 end-page: 2441 ident: RF7 article-title: On roots of random polynomials publication-title: Trans. Amer. Math. Soc. – year: 1984 ident: RF2 publication-title: Introduction to Multivariate Statistical Analysis – volume: 18 start-page: 439 year: 1968 ident: 10.1006/jmaa.2001.7554_RF4 article-title: The number of real zeros of a class of random algebraic polynomials publication-title: Proc. London Math. Soc. doi: 10.1112/plms/s3-18.3.439 – volume: 347 start-page: 4365 year: 1995 ident: 10.1006/jmaa.2001.7554_RF9 article-title: The complex zeros of random polynomials publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1995-1308023-8 – volume: 210 start-page: 724 year: 1997 ident: 10.1006/jmaa.2001.7554_RF5 article-title: Complex roots of a random algebraic polynomial publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1997.5429 – volume: 349 start-page: 2427 year: 1997 ident: 10.1006/jmaa.2001.7554_RF7 article-title: On roots of random polynomials publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-97-01766-2 – year: 1981 ident: 10.1006/jmaa.2001.7554_RF1 – year: 1998 ident: 10.1006/jmaa.2001.7554_RF6 – volume: 103 start-page: 1249 year: 1988 ident: 10.1006/jmaa.2001.7554_RF10 article-title: An asymptotic expansion for the expected number of real zeros of a random polynomial publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1988-0955018-1 – year: 1984 ident: 10.1006/jmaa.2001.7554_RF2 – year: 1986 ident: 10.1006/jmaa.2001.7554_RF3 – volume: 81 start-page: 207 year: 1979 ident: 10.1006/jmaa.2001.7554_RF8 article-title: On the average number of real zeros of a class of random algebraic curves publication-title: Pacific J. Math. doi: 10.2140/pjm.1979.81.207 |
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| SubjectTerms | Adler's theorem complex roots coordinate transform density of zeros Exact sciences and technology Jacobian of transformation Mathematical analysis Mathematics Number of complex zeros random algebraic polynomials random trigonometric polynomials real roots Sciences and techniques of general use Special functions |
| Title | Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients |
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