Covariance of the number of real zeros of a random trigonometric polynomial
For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ)=∑j=0n{ajcosjθ+bjsinjθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/3. However the variance of the abov...
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| Format: | Article |
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Wiley
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/28492 |
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| author | K. Farahmand M. Sambandham |
| author_facet | K. Farahmand M. Sambandham |
| author_sort | K. Farahmand |
| collection | DOAJ |
| description | For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ)=∑j=0n{ajcosjθ+bjsinjθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals (0,π) and (π,2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as Pn(θ)=∑j=0naj(ω)cosjθ, is also discussed. Tn(θ) has the advantage on Pn(θ) of being stationary, with respect to θ, for which, therefore, a more advanced method developed could be used to yield the results. |
| format | Article |
| id | doaj-art-a2daf605c4fc4977943e993c6e97592d |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-a2daf605c4fc4977943e993c6e97592d2025-02-03T01:29:14ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252006-01-01200610.1155/IJMMS/2006/2849228492Covariance of the number of real zeros of a random trigonometric polynomialK. Farahmand0M. Sambandham1Department of Mathematics, University of Ulster at Jordanstown, Co. Antrim BT37 0QB, United KingdomDepartment of Mathematics, Morehouse College, Atlanta, GA 30314, USAFor random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ)=∑j=0n{ajcosjθ+bjsinjθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals (0,π) and (π,2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as Pn(θ)=∑j=0naj(ω)cosjθ, is also discussed. Tn(θ) has the advantage on Pn(θ) of being stationary, with respect to θ, for which, therefore, a more advanced method developed could be used to yield the results.http://dx.doi.org/10.1155/IJMMS/2006/28492 |
| spellingShingle | K. Farahmand M. Sambandham Covariance of the number of real zeros of a random trigonometric polynomial International Journal of Mathematics and Mathematical Sciences |
| title | Covariance of the number of real zeros of a random trigonometric polynomial |
| title_full | Covariance of the number of real zeros of a random trigonometric polynomial |
| title_fullStr | Covariance of the number of real zeros of a random trigonometric polynomial |
| title_full_unstemmed | Covariance of the number of real zeros of a random trigonometric polynomial |
| title_short | Covariance of the number of real zeros of a random trigonometric polynomial |
| title_sort | covariance of the number of real zeros of a random trigonometric polynomial |
| url | http://dx.doi.org/10.1155/IJMMS/2006/28492 |
| work_keys_str_mv | AT kfarahmand covarianceofthenumberofrealzerosofarandomtrigonometricpolynomial AT msambandham covarianceofthenumberofrealzerosofarandomtrigonometricpolynomial |