This document presents research on the expected number of level crossings of random trigonometric polynomials. It begins with an abstract summarizing the background and objectives. The main results are then presented. It is shown that for random trigonometric polynomials of a certain form, with coefficients following a normal distribution, the expected number of real zeros in the interval from 0 to 2π is asymptotically equal to a particular expression involving n, as n becomes large. This expression is derived through applying the Kac-Rice formula and estimating a certain integral. The research builds on previous related works and clarifies results from one of those papers.

1. Journal for Research| Volume 01| Issue 12 | February 2016
ISSN: 2395-7549
All rights reserved by www.journalforresearch.org 37
Expected Number of Level Crossings of a Random
Trigonometric Polynomial
Dr. P. K. Mishra Dipty Rani Dhal
Associate Professor Research Scholar
Department of Mathematics Department of Mathematics
CET, BPUT, BBSR, Odisha, India “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 )=
   kba K
n
K
K
cos
1


. In the interval (Φ’, Φ’’). Assuming that ak(ω ) are independent random variables identically distributed
according to the normal law and that bk = kp
(p ≥ 0) are positive constants, we show that EN( T : 0, 2π ) ~
 nOn
p
p
n
log2)1(
32
12
2
1
2














 Outside an exceptional set of measure at most (2/ n ) where   
2
2
2
)(log'
32124
nSS
pp
n



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
T = Tn (Φ, ω) =    kba K
n
K
K
cos
1


( 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
  kgkgk kK
n
K
p
sincos 212
1


 (2)
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.
     )(23212
1
13
4
13
11
2
1 

q
nOnpp
Here, q= max(0, -p) and   .211321
q The measure of exceptional set does not exceed n-2 1
Das [2 ] took the polynomial (1) where aK(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) ]1/2
2n+O( n ) (3)
for bK = kp
( 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
EN (T; 0, 2π) ~   )(log21
32
12 2
1
2
nOn
p
p
n










Where
  
2
2
2
)(log'
32124
nSS
pp
n


 β = constant S ~ 1 S’ ~ 1
Our asymptotic estimate implies that Das’s estimate in [1] is approached from below. Also our term is smaller.

2. Expected Number of Level Crossings of a Random Trigonometric Polynomial
(J4R/ Volume 01 / Issue 12 / 007)
All rights reserved by www.journalforresearch.org 38
The particular case for p=0 has been considered by Dunnage [3] and Pratihari and Bhanja [4] . Dunnage has shown
that in the interval 0 ≤ θ ≤ 2π all save a certain exceptional set of the functions Tn( Φ, ω ) have zeros when n is large.
  




 13
3
13
11
log
3
2
nnO
n (4)
The measure of the exceptional set does not exceed ( logn )-1
. Using the kac_rice formula we tried to obtain in [4] that
Professor Dunnage[5] comments that our result is incorrect.
EN (T; 0, 2π) ~ )(log
6
2
nO
n

He is quite right when he says than an asymptotic estimate is unique and that both results (4) and (5) cannot be correct.
But in his calculations given in paragraph 4 of [5] he seems to have imported a factor 2 and the correct calculation would
give I’~ 2πn / √3. accepting his own statement in paragraph 3 that I < I’, our point is clear. However, since I’ ~ 2πn /√3
on direct integration, our estimation of EN as found in [4], contained in the statement (5) above, must be wrong. We are sorry
about our mistake. In this paper we consider our original integral I and evaluate it directly instead of placing it between two
integrals as in [4], the second one being possibly suspect . This rectification eventually raises our estimate for EN but, all the
same, keeps it below Dunnage’s estimate stated in above. The purport of our result is that EN approaches the value 2n/√3 from
below. This is something meaningful. We prove the following theorem.
Theorem
The average number of real zeros in the interval (0, 2π ) of the class of random trigonometric polynomials of the form
   kba K
n
K
K
cos
1


where aK( ω ) are mutually independent random variables identically distributed according to normal law with mean
zero and variance one and bK=kp
( p ≥ 0) are positive constants , is asymptotically equal to
  )(log21
32
12 2
1
2
nOn
p
p
n









 (5)
Outside an exceptional set of measure at most (2/n) where
  
2
2
2
)(log'
32124
nSS
pp
n


 β = constant S ~ 1 S’ ~ 1
II. THE APPROXIMATION FOR EN (T;0,2Π)
Let L(n) be a positive-valued function of n such that L(n) and n/L(n) both approach infinity with n. We take  =L(n)/n
throughout.
Outside a small exceptional set of ω, Tn( θ ,ω ) has a negligible number of zeros in each of the intervals (0, ), (π- ,π+
 ) and (2π- ,2π). By periodicity, of zeros in each of intervals (0, ) and (2π- ,2π) is the same as number in (- , ).
We shall use the following lemma, which is due to Das [2] .
Lemma
The probability that T has more than 1 + (log n)-1
(logn+logDn+4n ) Zeros in ω- ≤ θ ≤ ω+ does not exceed 2 exp(-n ),
where
Dn = 

n
K
K
b
1
The steps in this section follow closely those in section 2 of [4]. therefore, we indicate only the modifications necessary. In
this case we have
T=    kba K
n
K
K
cos
1


T’=    kbka K
n
K
K
sin
1



   






 

n
K
K
bkzkyzy
1
22
sincos
2
1
exp, 
 
 
    

 

 







 dybkzkkyzidzp
n
K
K
1
22
2
sincos
2
1
expexp
2
1
,0 
And finally

3. Expected Number of Level Crossings of a Random Trigonometric Polynomial
(J4R/ Volume 01 / Issue 12 / 007)
All rights reserved by www.journalforresearch.org 39
( 6)
for fixed non-zero real constants A and B to be chosen .
III.ESTIMATION OF THE INTEGRAL OF EQUATION (6)
Consider the integral
 
 
du
BAu
bkkku
I
n
K
K




















 2
1
22
sincos
log

Which exists in general as a principal value if
A2
= 

n
K 1
bK
2
cos2
kθ
Let
B2
=


n
K 1
k2
bK
2
sin2
kθ
And
C2
=


n
K 1
kbk
2
coskθ sinkθ
As in Das[2], letting bk = kp
(p ≥ 0) we get
)()(
2222
2
nL
n
nL
n
OC
pp 










Say (β = constant)
outside the set {0,  π,  2π, … } of the values of θ, AB > C2
. We have
 
 
du
BAu
bkkku
I
n
K
K




















 2
1
22
sincos
log

du
bbuu
bucu


 







 22
222
2
2
log
Where c=(C/A) and b=(B/A). Now by integration by parts,
     du
bucu
ucux
bucuudubucu
X X
  


0 0
222
22
222222
2
22
0
2log2log = 








  X
Odu
bucu
buc
XcXX
X
1
2
222log2
0
222
22
2
And
    


0
0
222222
2log2log
X
X
dubucudubucu = 








  X
Odu
bucu
buc
XcXX
X
1
2
222log2
0
222
22
2
Therefore
  








 
 
X
Odu
bucu
buc
XXXdubucu
X
X
X
X
1
2
24log42log 222
22
222
Again
  








 
 
X
Odu
bbuu
bbu
XXXdubbuu
X
X
X
X
1
2
24log42log 22
2
22
Hence the integral

4. Expected Number of Level Crossings of a Random Trigonometric Polynomial
(J4R/ Volume 01 / Issue 12 / 007)
All rights reserved by www.journalforresearch.org 40











du
bbuu
bucu
X
X
22
222
2
2
log du
bucu
buc
X
X




222
22
2
2 




X
X
bbuu
bbu
22
2
2
2 






X
O
1
  
x
x
cb
cu
cb
x
x
bucucdu
bucu
buc
X
X













 

 42
2
1422222
222
22
tan22log
2
2






















22
42
142
222
222
2 2
tan2
2
2
log
Xb
cbX
cb
bXcX
bXcX
c
When X→∞, we have
42
222
22
2
cbdu
bucu
buc












0
2
22
2
du
bbuu
bbu
and
O 





X
1 =0
Therefore
I =

X
X
x
lim . . . = du
bbuu
bucu


 







22
222
2
2
log
 
2
1
4
422
2
1
42
22 






 

A
CBA
cb
  n
p
p
S
S
n





















 21
32
12'
2
1
2
Where
  
2
2
2
))(('
32124
nLSS
pp
n


 S ~ 1 S’ ~ 1
Thus
       2
1
2
2
1
123212~ n
nppI 
Now the result follows from the section 4 of [4] , choosing L(n) =log n. the cases where -1/2 < p < 0 and p= -1/2 can
be similarly dealt with and results can be obtained to show that Das’s estimates are approached from below.
We express our reverence and deep gratitude to Sir A.C.offord for his approval of the thesis [6] which presents this
result, and for his valuable suggestions .
REFERENCES
[1] Das, M.K. Real zeros of a random sum of orthogonal polynomials, Proc. Amer. Math. Soc. 27, 1 (1971), 147-153.
[2] Das, M.K. Real zeros of a class of random algebraic Polynomials, Journal of Indian Math. Soc.. 36 (1972), 53-63.
[3] Dunnage, J.E.A. The number of real zeros of a class of random algebraic polynomials (I), Proc. London Math. Soc. (3) 18 (1968), 439-460.
[4] Pratihari and Bhanja .On the number of real zeros of random trigonometric polynomial, Trans. Of Amer. Math. Soc. 238 (1978) 57-70.