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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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 262; no. 2; pp. 554 - 563
Main Authors: Farahmand, K., Grigorash, A.
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 15.10.2001
Elsevier
Subjects:
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
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
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary: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.
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.2001.7554