International Journal of Computational Science and Mathematics.
ISSN 0974-3189 Volume 7, Number 1 (2015), pp. 19-24
© International Research Publication House
http://www.irphouse.com
Number Of Real Zeros Of Random Trigonometric
Polynomial
1
Dr.P.K.Mishra, 2DR.A.K.Mahapatra , 3Dipty Rani Dhal
Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
& 3. Department of Mathematics, ITER, SOA,BBSR, ODISHA, INDIA
1
[email protected], [email protected]
Corresponding Author’s Email: [email protected]
DEPARTMENT OF MATHEMATICS, CET, BPUT, BBSR, ODISHA
DEPARTMENT OF MATHEMATICS, ITER, SOA , BBSR, ODISHA
1.
2.
1991 Mathematics subject classification (Amer. Math. Soc.): 60 B 99.
Keywords and phrases: Independent identically distributed random variables,
random algebraic polynomial, random algebraic equation, real roots.
THEOREM:-Let EN (T, Φ’, Φ’’) denote the average number of real zeros of the
random trigonometric polynomial
n
T T n (θ, ω)   a k (ω) b k cos kθ
k 1
in the interval (Φ’, Φ’’). Assuming ak(w) are independent random variables
identically distributed according to the normal law that b k  k p ( p  0) are positive
constants we show that
1
2
 2p  1 

EN( T;0,2π ) ~ 
(1  ε n 2  2n  0(log n ) .
 2p  3 

Outside an exceptional set of measure at most (2/n) where
ε2 n 
4β 2 ( 2p  1)(2p  3)
SS(log n ) 2
β=constant
s~1
S’~1
1. INTRODUCTION:- Let N (T, Φ’, Φ’’) be the number of real roots of the
trigonometric polynomial
20
Dr.P.K.Mishra et al
n
T T n (θ, ω)   a k (ω) b k cos kθ (1)
k 1
in the interval (Φ’, Φ’’) where the coefficients ak(w) are mutually independent
random variables identically distributed according to the normal law, bk=kp are
positive constants and when multiple roots are counted only once. Let EN (T, Φ’, Φ’’)
denote the exception of N (T, Φ’, Φ’’). Obviously, Tn(θ, w) can have at most 2n zeros
in the interval (0, 2π). Das [1] studied the class of polynomials
n
k
p
(g 2k 1 cos kθ g 2 k 1 sin kθ)
(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
 11  4q 13 ε 
2
1
[(2p  1)/(2p  3)] 2n  O n 13



number of real roots when n is large. Here, q= max (0, -p) and ε 1  (2 / 13)(1  2q).
The measure of the exceptional set does not exceed n 2 ε .
1
1
Das [2] took the polynomial (1) where ak(w) are independent normal random
variables identically distributed with mean zero and variance done. He proves that in
the interval 0  θ  2 π , average number of real zeros of polynomials (1) is
1
[(2p  1)/(2p  3)] 2 2n  On 
(3)
1

For b k  k p  p   and of the order of n p 3 2 if p  -1 2 for large n.
2

In this paper we consider the polynomial (1) with conditions as in das [2] and
use the Kac-Rice formula for the exception of the number of real zeros and obtain that
for p≥0.
Where
1
2
 2p  1 

EN( T;0,2π ) ~ 
(1  ε n 2  2n  0(log n )
 2p  3 

2
4β ( 2p  1)(2p  3)
ε2 n 
SS(log n ) 2
β=constant
S~1
S’~1
Our asymptotic estimate implies that Das’s estimate in [1] is approached from
below.Also our error term is smaller.
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  2 π all save a certain
exceptional set of the functions (Tn(θ,w) have
2n
 O ( n 11
3
13
(log n ) 3 13 )
(4)
Number Of Real Zeros Of Random Trigonometric Polynomial
21
zeros when n is large. The measure of the exceptional set does not exceed (log n)-1.
Using the Kac-Rice formula we tried to obtain in [4] that
 2n 
EN( T;0,2π ) ~ 
  0(log n )
 6 
(5)
Professor Dunnage [5] comments that our result is incorrect. He is quite right
when he says than an asymptotic estimate is unique and both results (4) and (5)
cannot be correct. But in his calculations given in paragraph 4 and 5 he seems to have
imported a factor 2 and the correct calculation would give I ~ 2 πn / 3 . Accepting
his own statement in paragraph 3 that I<I’, our point is clear. However, since
I ~ 2 πn / 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 (4) above. The purpose 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
n
a
k
(ω)b k cos kθ
k 1
where ak(w) are mutually independent random variables identically distributed
according to the normal law with mean zero and variance one and bk=kp(p≥0) are
positive constants, is asymptotically equal to
1
 2p  1 
 2
2
(1 ε n  2n  0(log n )

 2p  3 

outside an exceptional set of measure at most (2/n) where
4β 2 ( 2p  1)(2p  3)
ε n
SS(log n ) 2
2
β=constant
S~1
S’~1
2. 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
ε  L(n ) / n throughout.
Outside a small exceptional set of values of w, (Tn(θw) has a negligible number
of zeros in each of the intervals (0, ε ), (π  ε, π  ε ) and (2π  ε, 2π) . By periodicity,
the number of zeros in (0, ε ) and (2π  ε, 2π) is the same as the number in (ε, ε) .
We shall use the following lemma, which is due to Das[2].
22
Dr.P.K.Mishra et al
LEMMA :-
The
probability
that
T
has
more
than
1  (log n ) (logn  log D n 4nε) zeros in          does not exceed 2 exp
1
n
(-nε), where D n   b k
k 1
The step in this section follow closely those in section 2 of 4. Therefore, we
indicate only the modifications necessary. In this case we have
n
n
T   a k (ω) b k cos kθ
T   - ka k (ω) b k sin kθ
k 1
k 1
n
 1
2
φ( y, z )  exp   ( y cos kθ  zk sin kθ) 2 b k 
 2 k 1



1
 1 n
2
2
p(0, η) 
dz
exp(
i
η
z
)
exp
  ( y cos kθ - zk sin kθ) b k dy

2 

(2 π) 
 2 k 1

and finally
2 
 n
( y cos kθ - zk sin kθ) 2 b k 



1 
 k 1

du
 η p(0, η)dn  (2π) 2  log
2
(
Au

B
)




(6).
for fixed non-zero real constants A and B to be chosen.
3. ESTIMATION OF THE INTEGRAL OF EQUATION :-Consider the integral
 n
2 
(u
Cos
k
θ

k
Sin
k
θ
)




I   log  k 1
du
2
(
Au

B
)




n
which exists in general as a principal value if ,
A 2   cos 2 kθ
k 1
Let B2=
n
n
k
1
2
sin 2 kθ and C 2   k cos kθ sin kθ.
1
As in Das[2] letting bk=kp(p≥0) we get
A2 
 1  1 n 2 p1
1 n 2 p 1 
1

0

  S

2 2p  1 
 L( n )   2 2 p  1
 1   1 n 2 p 3
1 n 2 p 3 
  S'
B 
1  0
2 2p  3 
 L( n )   2 2 p  3
2
 n 2 p  2  βn 2 p  2
 
C  0
L
(
n
)

 L( n )
2
( β  constant)
say,
say,
Number Of Real Zeros Of Random Trigonometric Polynomial
23
Outside the set 0, π ,  2 π of the values of θ, AB>C2. We have
 n

(u cos kθ  k sin kθ) 2 b 2 k 

 

I   log  k 1
du
2
(
Au

B
)






 u 2  2c 2 u  b 2 
=  log  2
du
2 

 u  2bu  b 
where c= (C/A) and b=(B/A). Now by integration by parts,
x x  2 u 2  2c 2 u 
2
0 log(u  2c u  b )du  [ulog(u  2c u  b )]0    u 2  2c 2 u  b 2 du
x
2
2
2
2
2
c 2 u  b2
1
du  0 
2  2c 2 u  b 2
X
0 u
x
= 2 X log X  2c 2  2 X  2 
and
0
x
2
2
2
2
2
2
 log(u  2c u  b )du   log(u  2c u  b )du
x
0
 c 2 u  b2
1
du  0 
2  2c 2 u  b 2
X
0 u
x
= 2 X log X  2c 2  2 X  2 
Therefore
c2u  b2
1
du  0 
2  2c 2 u  b 2
X
x u
x
x
2
2
2
 log( u  2c u  b )du  4X log X  4X  2 
x
Again
x
bu  b 2
x
1
du  0 
2  2bu  b 2
X
x u
2
2
 log(u  2bu  b )du  4X log X  4X  2 
x
Hence the integral
x
x
 u 2  2c2 u  b 2 
c2u  b 2
bu  b2
1
log


du

2
du

2
du

0
 
  u2  2bu  b2 
 2 2 2
 2
2
 X
x
x u  2c u  b
x u  2bu  b
x
Obviously
x
x
2
2
 c u b 
x
2  2
du c2 logu2  2c2u  b2 x 2
2
2
x u  2c u  b 
 1 u2  2c2u b2 
b c  tan

b2 c4 

 x
2
2
4 

 X2  2c2X b2 
2
4
1 2X  b  c  
c2 log 2


2

b

c

tan
2
2
 b2  X2 
 X  2c X b 


When X→∞, we have
4


24
Dr.P.K.Mishra et al


   u
2


   u
And
0(
2

c2u  b 2
 du  π
2
2 
 2c u  b 
bu  b 2
 2 bu  b 2
b
2
 c4


 du  0

1
)0
X
Therefore


 u2  2c2u  b2 
 A2 B2  C4 
2
4 1/ 2


(7)
I  lim......... log. 2
du

2

(
b

c
)

2

2 
A4
x 
 u  2bu  b 



 S '  2 p  1
= 
 
S

 2 p  3

where
2n 


 (1   2 n )  2  n


4  2 ( 2 p  1 )( 2 p  3 )
2
SS ' L ( n ) 
S ~1
S' ' ~ 1
Thus

I ~ (2p  1)/(2p  3) 1/2 2  n (1   2 n )

1/2
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 estimate are approached .
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
DAS, M., 1972, Math. Student, A., 40, 305.
DAS, M., 1968, Proc. Comb. Phil. Soc., 64, 721.
DUNNAGE, J.E.A., 1966, Proc. London math. Soc., 16, 53.
PRATIHARI, D., AND BHANJA, M., 1982, Int. J. Math, Educ, Sci, Technol.,
13(1), 87.
DUNNAGE, J.E.A., 1983, MathematicalRev., 83, 60047
BHANJA, M. 1986, Ph.D Thesis, Utkal University.