Journal for Research| Volume 01| Issue 12 | February 2016 ISSN: 2395-7549
Expected Number of Level Crossings of a Random Trigonometric Polynomial Dr. P. K. Mishra Associate Professor Department of Mathematics CET, BPUT, BBSR, Odisha, India
Dipty Rani Dhal Research Scholar Department of Mathematics “SOA" University, Bhubaneswar
Abstract Let EN( T; Φ’ , Φ’’ ) denote the average number of real zeros of the random trigonometric polynomial T=Tn( Φ, ω )= n
a b K
K
cos k
. In the interval (Φ’, Φ’’). Assuming that ak(ω ) are independent random variables identically distributed
K 1
according to the normal law and that bk = kp (p ≥ 0) are positive constants, we show that EN( T : 0, 2π ) ~ Outside an exceptional set of measure at most (2/ n ) where 2 4 2 2 p 12 p 3 2 p 1 1
2
(1 n ) 2 p 3 2
2 n O log n
n
SS ' (log n )
2
β = constant S ~ 1, S’ ~ 1. 1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99. Keywords: Independent, Identically Distributed Random Variables, Random Algebraic Polynomial, Random Algebraic Equation, Real Roots, Domain of Attraction of the Normal Law, Slowly Varying Function _______________________________________________________________________________________________________
I. INTRODUCTION Let N( T ; Φ’ , Φ’’ ) be the number of real zeros of trigonometric polynomial n
T = Tn (Φ, ω) = a K b K cos k
(1)
K 1
In the interval ( Φ’ , Φ’’ ) where the coefficients ak(ω) are mutually independent random variables identically distributed according to the normal law; bk=kp are positive constants and when multiple zeros are counted only once . Let EN (T; Φ’, Φ’’) denote the expectation of N (T; Φ’, Φ’’). Obviously, Tn (Φ, ω) can have at most 2n most zeros in the interval (0, 2π) Das [ 1 ] studied the class of polynomials n
k g p
2 K 1
cos k g 2 k sin k
(2)
K 1
where gk are independent normal random variables for fixed p > -1/2 and proved that in the interval (0 , 2π) the function (2) have number of real roots when n is large.
2 p 1 2 p 3
11
1
4q
1
2 n O ( n 13 13 ) Here, q= max(0, -p) and 1 2 13 1 2 q . The measure of exceptional set does not exceed n-2 1 Das [2 ] took the polynomial (1) where a K(ω ) are independent normal random variables identically distributed with mean zero and variance one . He proves that in the interval 0 ≤ θ ≤ 2π , the average number of real zeros of polynomials ( 1 ) is [(2p+1) / (2p+3) ]1/22n+O( n ) (3) p for bK = k ( p > -1/2 ) and of the order of np+3/2 if -3/2 ≤ p ≤ -1/2 for large n. In this paper we consider the polynomial (1) with conditions as in DAS [2] and use the Kac_Rice formula for the expectation of the number of real zeros and obtain that for p ≥ 0 2
EN (T; 0, 2π) ~ 2 p 1 1 2 n 2 p 3
1 2
2 n O (log n )
Where 2
n
4
2
2 p 12 p 3 SS ' (log n )
β = constant S ~ 1
S’ ~ 1
2
Our asymptotic estimate implies that Das’s estimate in [1] is approached from below. Also our term is smaller.
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