Number Of Real Zeros Of Random Trigonometric Polynomial

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1 Iteratioal Joral of Comtatioal iee ad Mathematis. IN Volme 7, Nmer (5),. 9- Iteratioal Researh Pliatio Hose htt:// Nmer Of Real Zeros Of Radom Trigoometri Polyomial Dr.P.K.Mishra, DR.A.K.Mahaatra, 3 Dity Rai Dhal. Deartmet of Mathematis, CET, BPUT, BBR, ODIHA, INDIA. & 3. Deartmet of Mathematis, ITER, OA,BBR, ODIHA, INDIA mishradr@gmail.om, ditydhal@gmail.om Corresodig Athor s ditydhal@gmail.om DEPARTMENT OF MATHEMATIC, CET, BPUT, BBR, ODIHA DEPARTMENT OF MATHEMATIC, ITER, OA, BBR, ODIHA 99 Mathematis sjet lassifiatio (Amer. Math. o.): 6 B 99. Keywords ad hrases: Ideedet idetially distrited radom variales, radom algerai olyomial, radom algerai eqatio, real roots. THEOREM:-Let EN (T, Φ, Φ ) deote the average mer of real zeros of the radom trigoometri olyomial T T (θ,ω) a (ω) os θ i the iterval (Φ, Φ ). Assmig a (w) are ideedet radom variales idetially distrited aordig to the ormal law that ( ) are ositive ostats we show that EN( T;,π ) ~ ( ε (log ). 3 Otside a eetioal set of measre at most (/) where β ( )( 3) ε (log ) β=ostat s~ ~. INTRODUCTION:- Let N (T, Φ, Φ ) e the mer of real roots of the trigoometri olyomial

2 Dr.P.K.Mishra et al T T (θ,ω) a (ω) os θ () i the iterval (Φ, Φ ) where the oeffiiets a (w) are mtally ideedet radom variales idetially distrited aordig to the ormal law, = are ositive ostats ad whe mltile roots are oted oly oe. Let EN (T, Φ, Φ ) deote the eetio of N (T, Φ, Φ ). Oviosly, T (θ, w) a have at most zeros i the iterval (, π). Das [] stdied the lass of olyomials (g osθ g si θ) () where g are ideedet ormal radom variales for fied >-/ ad roved that i the iterval (, π) the ftio () have q 3 ε [( )/( 3)] O 3 mer of real roots whe is large. Here, q= ma (, -) ad (/3)( q). ε The measre of the eetioal set does ot eeed. Das [] too the olyomial () where a (w) are ideedet ormal radom variales idetially distrited with mea zero ad variae doe. He roves that i the iterval θ π, average mer of real zeros of olyomials () is [( )/( 3)] O ε (3) 3 For ad of the order of if - for large. I this aer we osider the olyomial () with oditios as i das [] ad se the Ka-Rie formla for the eetio of the mer of real zeros ad otai that for. EN( T;,π ) ~ ( ε (log ) 3 β ( )( 3) Where ε (log ) β=ostat ~ ~ Or asymtoti estimate imlies that Das s estimate i [] is aroahed from elow.also or error term is smaller. The artilar ase for = has ee osidered y Dage [3] ad Pratihari ad Bhaja []. Dage has show that i the iterval θ π all save a ertai eetioal set of the ftios (T (θ,w) have O ( (log ) ) () 3

3 Nmer Of Real Zeros Of Radom Trigoometri Polyomial zeros whe is large. The measre of the eetioal set does ot eeed (log ) -. Usig the Ka-Rie formla we tried to otai i [] that EN( T;,π ) ~ (log ) (5) 6 Professor Dage [5] ommets that or reslt is iorret. He is qite right whe he says tha a asymtoti estimate is iqe ad oth reslts () ad (5) aot e orret. Bt i his allatios give i aragrah ad 5 he seems to have imorted a fator ad the orret allatio wold give I ~ π / 3. Aetig his ow statemet i aragrah 3 that I<I, or oit is lear. However, sie I ~ π / 3 o diret itegratio, or estimatio of EN as fod i [], otaied i the statemet (5) aove, mst e wrog. We are sorry aot or mistae. I this aer we osider or origial itegral I ad evalate it diretly istead of laig it etwee two itegrals as i [], the seod oe eig ossily sset. This retifiatio evetally raises or estimate for EN t, all the same, ees it elow Dage s estimate stated i () aove. The rose of or reslt is that EN aroahes the vale / 3 from elow. This is somethig meaigfl. We rove the followig theorem. Theorem. The average mer of real zeros i the iterval (, π) of the lass of radom trigoometri olyomials of the form a (ω) osθ where a (w) are mtally ideedet radom variales idetially distrited aordig to the ormal law with mea zero ad variae oe ad = ( ) are ositive ostats, is asymtotially eqal to ( ε (log ) 3 otside a eetioal set of measre at most (/) where β ( )( 3) ε (log ) β=ostat ~ ~. THE APPROIMATION FOR EN( T;,π ) Let L() e a ositive-valed ftio of sh that L() ad /L() oth aroah ifiity with. We tae ε L()/ throghot. Otside a small eetioal set of vales of w, (T (θw) has a egligile mer of zeros i eah of the itervals (,ε),(π ε, π ε) ad (π ε, π). By eriodiity, the mer of zeros i (,ε) ad (π ε, π) is the same as the mer i ( ε,ε). We shall se the followig lemma, whih is de to Das[].

4 Dr.P.K.Mishra et al LEMMA :- The roaility that T has more tha (log ) (log logd ε) zeros i does ot eeed e (-ε), where D The ste i this setio follow losely those i setio of. Therefore, we idiate oly the modifiatios eessary. I this ase we have T a (ω) os θ T - a (ω) si θ φ(y,z) e (yosθ zsi θ) (, η) dz e(iηz) e (yos θ si θ) (π) - z dy ad fially (yos θ - z si θ) log d (6). η (,η)d (π) (A B) for fied o-zero real ostats A ad B to e hose. 3. ETIMATION OF THE INTEGRAL OF EQUATION :-Cosider the itegral ( Cos θ i θ) I log d (A B) whih eists i geeral as a riial vale if, Let B = si θ ad C os θ si θ. As i Das[] lettig = ( ) we get A L() B C 3 3 L() β L() L() ' A 3 3 os ( β ostat) θ say, say,

5 Nmer Of Real Zeros Of Radom Trigoometri Polyomial 3 Otside the set,, π π of the vales of θ, AB>C. We have d B) (A si θ) ( osθ log I = d log where = (C/A) ad =(B/A). Now y itegratio y arts, )] [ )d d = d log ad )d )d = d log Therefore d log )d Agai d log )d Hee the itegral d d d log Oviosly ta log ta log d Whe, we have

6 Dr.P.K.Mishra et al d π d Ad ( ) Therefore A B C I lim... log. ( ) / d A (7) = where Ths ' ( I ' ~ ( 3 L ( ) )( ) 3) / ( )/( 3) ( ) / ~ ' ' ~ Now the reslt follows from the setio of [], hoosig L()=log. The ases where /< < ad =-/ a e similarly dealt with ad reslts a e otaied to show that Das s estimate are aroahed. REFERENCE [] DA, M., 97, Math. tdet, A.,, 35. [] DA, M., 968, Pro. Com. Phil. o., 6, 7. [3] DUNNAGE, J.E.A., 966, Pro. Lodo math. o., 6, 53. [] PRATIHARI, D., AND BHANJA, M., 98, It. J. Math, Ed, i, Tehol., 3(), 87. [5] DUNNAGE, J.E.A., 983, MathematialRev., 83, 67 [6] BHANJA, M. 986, Ph.D Thesis, Utal Uiversity.