scholarly journals On random orthogonal polynomials

2001 ◽  
Vol 14 (3) ◽  
pp. 265-274 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Dependent Random Variables ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than −u. Promoted by the graphical interpretation, we define these crossings as u-sharp. For the above polynomial, we provide the expected number of such crossings.

scholarly journals Real zeros of a random polynomial with Legendre elements

1997 ◽  
Vol 10 (3) ◽  
pp. 257-264
Author(s):  
K. Farahmand
Keyword(s):  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables with mean zero and variance one is known. The present paper considers the case when the means and variances of the coefficients are not all necessarily equal. It is shown that in general this expected number of real zeros is only dependent on variances and is independent of the means.

scholarly journals On real zeros of random polynomials with hyperbolic elements

1998 ◽  
Vol 21 (2) ◽  
pp. 347-350
Author(s):  
K. Farahmand ◽  
M. Jahangiri
Keyword(s):  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomials ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomialg1coshx+2g2cosh2x+…+ngncoshnxwheregj,(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently largenthis asymptotic value is(1/π)logn.

scholarly journals Zeros of an algebraic polynomial with nonequal means random coefficients

2004 ◽  
Vol 2004 (63) ◽  
pp. 3389-3395
Author(s):  
K. Farahmand ◽  
P. Flood
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomiala0+a1x+a2x2+⋯+an−1xn−1. The coefficientsaj(j=0,1,2,…,n−1)are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval(−1,1)and the mean of the last group in(−∞,−1)∪(1,∞).

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals On the variance of the number of real zeros of a random trigonometric polynomial

1997 ◽  
Vol 10 (1) ◽  
pp. 57-66 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
General Formula ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Normal Process ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Upper Estimate

The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ+g2cos2θ+…+gncosnθ where gj(j=1,2,…,n) is a sequence of independent normally distributed random variables is known. The present paper provides an upper estimate for the variance of such a number. To achieve this result we first present a general formula for the covariance of the number of real zeros of any normal process, ξ(t), occurring in any two disjoint intervals. A formula for the variance of the number of real zeros of ξ(t) follows from this result.

scholarly journals Real zeros of random algebraic polynomials with binomial elements

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
A. Nezakati ◽  
K. Farahmand
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
General Theorem ◽  
Random Variables ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials ◽  
Special Case

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1) are assumed to be independent normal random variables with mean zero. For integers m and k=O(log⁡n)2 the variances of the coefficients are assumed to have nonidentical value var⁡(aj)=(k−1j−ik), where n=k⋅m and i=0,1,2,…,m−1. Previous results are mainly for identically distributed coefficients or when var⁡(aj)=(nj). We show that the latter is a special case of our general theorem.

scholarly journals Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one, is(1/π)logn. This result is true for allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

scholarly journals Covariance of the number of real zeros of a random trigonometric polynomial

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Trigonometric Polynomial ◽  
Random Coefficients ◽  
Expected Number ◽  
Asymptotic Value ◽  
Advanced Method ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial

For random coefficientsajandbjwe consider a random trigonometric polynomial defined asTn(θ)=∑j=0n{ajcos⁡jθ+bjsin⁡jθ}. The expected number of real zeros ofTn(θ)in the interval(0,2π)can be easily obtained. In this note we show that this number is in factn/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 asPn(θ)=∑j=0naj(ω)cos⁡jθ, is also discussed.Tn(θ)has the advantage onPn(θ)of being stationary, with respect toθ, for which, therefore, a more advanced method developed could be used to yield the results.

scholarly journals Algebraic polynomials with random coefficients

2002 ◽  
Vol 15 (1) ◽  
pp. 83-88
Author(s):  
K. Farahmand
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Asymptotic Value ◽  
Fixed Probability ◽  
Special Case ◽  
Normally Distributed

This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω)+a1(ω)(n1)1/2x+a2(ω)(n2)1/2x2+…an(ω)(nn)1/2xn when n is large. The coefficients {aj(w)}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr). A special case of dependent coefficients is also studied.

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