Expected Number of Level Crossings of Legendre Polynomials

Size: px

Start display at page:

Download "Expected Number of Level Crossings of Legendre Polynomials"

Peregrine Pitts
5 years ago
Views:

1 Expected Number of Level Crossigs of Legedre olomials ROUT, LMNAYAK, SMOHANTY, SATTANAIK,NC OJHA,DRKMISHRA Research Scholar, G DEARTMENT OF MATHAMATICS,COLLEGE OF ENGINEERING AND TECHNOLOGY,BHUBANESWAR,ODISHA Research Scholar, G DEARTMENT OF MATHAMATICS,COLLEGE OF ENGINEERING AND TECHNOLOGY,BHUBANESWAR,ODISHA Research Scholar, G DEARTMENT OF MATHAMATICS,COLLEGE OF ENGINEERING AND TECHNOLOGY,BHUBANESWAR,ODISHA Research Scholar, G DEARTMENT OF MATHAMATICS,COLLEGE OF ENGINEERING AND TECHNOLOGY,BHUBANESWAR,ODISHA Facult, COLLEGE OF ENGINEERING AND TECHNOLOGY, BHUBANESWAR,ODISHA Facult, COLLEGE OF ENGINEERING AND TECHNOLOGY, BHUBANESWAR,ODISHA ABSTRACT:-The aim of this paper is to estimate the umber of real zeros of a orthogoal radom polomial uder differet coditio whe the coefficiets belog to the domai of attractio of orthogoal properties Let 0 w be radom polomial such that [ 0 w, w, w] is a sequece of mutuall idepedet, ormall distributed radom variables with mea zero ad variace uit ad [Ψ 0, Ψ ] be a sequece of ormalized orthogoal Legedre polomials, defied b t, where is the classical Legedre polomial The, for a costat K such that K / 0 as, the mathematical expectatio of umber of real zeros of the equatio w = is asmptotic to / 0 I Basic idea about o of zeros or o of level crossig A Level crossig: i geeral a level crossig meas crossig of a railwa or a railroad crossig is a place where a lie ad a road itersect each other o the same leveli mathematics o of level crossig meas o of zeros or poit of itersectio betwee a curve ad x-axisfor example: the trigoometric fuctios si x ad cos x the poit of itersectio of the curve with x axis are show i the followig figures called umber of zeros or umber of level Copright to IJARSET wwwijarsetcom 5

2 crossigs of the above two curves with x-axis Figure :- Here umber of zeros for si x is π ad o of zeros for cos x is +π/ for is a iteger Orthogoal polomial: A orthogoal polomial is a famil of polomial such that a two differet polomial i the sequece are orthogoal to each other uder some ier product The most widel used orthogoal polomials are classical orthogoal polomial cosistig ofa Legedre polomial, b Hermite polomial, c Jacobi polomial d Bessel polomial Legedre polomial: The solutio Legedre differetial equatio are a set of fuctios ow as the Legedre polomials The polomials are defied o[-,] We ca call Legedre polomials i Mathematics usig: Legedre[,x] Where represet the polomial, ad x is the variable Solutio to differetial equatio: Legedre polomial are oe of the solutios to the Legedre differetial equatio ; x " x + + = 0 Here is a costat is ow as Legedre differetial equatio To obtai the solutio of we shall use series solutio method A expressio for the Legedre polomial x is give b the formula ow as Rodrigues s formula of degree x = d! dx x Recursive formula for Legedre polomial: a d dx x x d dx x = x bx d dx x d dx x = x c + + x + x x + x = 0 d d dx x d dx x = + x Copright to IJARSET wwwijarsetcom 54

3 Legedre polomials are defied to be orthoormal, meaig the itegral of a product of Legedre polomials is either zero or oe I other words there is either a orthoormal costat N st N x x dx = The are orthogoal i[-,] x m x dx = 0 for m ot eqaul to x dx = + Zeros of Legedre polomial: The Legedre polomials are For = the Legedre polomial is 0 x =, x = x, x= + x, x = x + 5x ad so o we ow that the eve order Legedrepolomial are eve ad odd orders are odd fuctio Accordig to the geeral result about the zeros of solutios, the th polomial should have zeros i the iterval[-,] Let 0 w IITheorem be radom polomial such that [ 0 w, w, w] is a sequece of mutuall idepedet, ormall distributed radom variables with mea zero ad variace uit ad [Ψ 0, Ψ ] be a sequece of ormalized orthogoal Legedre polomials, defied b t, where is the Copright to IJARSET wwwijarsetcom 55

4 classical Legedre polomial The, for a costat K such that K / 0 as, the mathematical expectatio of umber of real zeros of the equatio w = is asmptotic to / 0 Let f f t,w 0 w IIIINTRODUCTION Where { 0, w, w, w} is a sequece of idepedet radom variables o a probabilit space Ω, A,, each ormall distributed with mathematical expectatio zero ad variace oe Let {Ψ 0, Ψ } be a sequece of ormalized orthogoal Legedre polomials Let ENf: α,β be the expected umber of real zeros of the equatio f= i the iterval α t β, where multiple zeros are couted ol oce We ow, from the wor of Das [], that i the iterval st t, all save a certai exceptioal set, the fuctios f have 0 zeros, o the average, whe is large ad =0 The measure of the exceptioal set does ot exceed exp -/ Farahmad [] has show that whe Ψ =t, has excepted umber of -level crossigs of the algebraic polomial Q 0 w t satisfies EN Q: -,= /π log /K, EN Q: -,-= EN Q:, ~ π - log A correspodig estimate, whe Ψ =cost, is also due to Farahamad [4] who showed that the radom trigoometric polomial as K / 0 as T w cos t has 0 0 expected umber of real zeros as log Compariso of these results with our theorem shows the differece ad similarit of behaviour of algebraic ad trigoometric polomial with the orthogoal polomial cosidered b us Thus, the umber of crossigs of the algebraic polomial with the level K decreases as K icreases, while for trigoometric polomial ad orthogoal polomial f the average umber of level crossigs remais fixed with probabilit oe, as log as K / 0 as I order to estimate EN f: -, for the polomial f w K, 0 To fid umber of level crossigs of the above polomial i the iterval -, we divide the iterval -, ito 7 subitervals -, -+, -+, - ad -, as i the, where 0 First, we derive umber of level crossigs of the above polomial i the iterval -+, - usig a well ow formula to fid the umber of level crossig of a polomial called Kac-Rice formula, we use its exteded formula for our use IV Exteded Kac-Rice Formula for EN f: α,β From Das[] ad Crammerwe fid that the expected umber of real zeros of the equatio f=0 i the iterval α,β satisfies: 4, Copright to IJARSET wwwijarsetcom 56

5 EN f :, dt x d, where Φx, is the desit fuctio of the distributio of f ad its derivative f Let X X Y Y 0, 0 Z Z ad 0 X Z Y The, the joit desit of f,f is x, π Let S X / From, we have exp - Zx Yx X / 4 YKs x, d / xexp -Z / s exp ds 5 X uttig p= YKs X i 5, we have Where θ p x, d s{exp ps ps}exp -s ds 0 = θ{p} θ p, s{expps s ds = p/ expp / 4 exp{ s p/ ds 6 p/ expp / 4{ π/ erfp/ Where f x= 0 exp t dt 0 Copright to IJARSET wwwijarsetcom 57

6 Hece from, 4, 5 ad 6 β EN f : α,β exp Z / πx β α / π α KY exp / X dt / / Xerf KY / X dt =I α,β+ I α,β 7 we divide the iterval -, ito subitervals -, -+, -+, - ad -, Combiig we prove theorem V Average Number of Level Crossigs i the Iterval -+ε, - ε The ChristoffelDarboux formula, Sasoe [5] for Legedre polomials reads as follows: 0 u u u u t 8 uttig h =+ - ad μ =+ ad followig the procedure described i last sectio of the paper, we have X 0 9 Y 0 0 For Legedre polomial, we have the relatios t t Ad t t From ad, we obtai t = Ad t = t 4 Differetiatig ad 4, ad multiplig them respectivel b ad + we obtai, after simplificatios Copright to IJARSET wwwijarsetcom 58

7 t 6t - t 6 t FromSasoe [5] we have t t 6 ad t t 7 Hece, usig 6 ad 7, we have t t I the rage ε γ π ε ad 0 < ε< π/, the asmptotic estimate of, for t=cos γ, is give b Sasoe [5] 5 8 Hece π t - / πsi γ t O π cos γ Osi γ 4 π π cos γ cos γ π si γ 4 4 π π cos γ cos γ cos γ O 4 4 From 7 ad 9, we obtai t π O t t / B the first theorem of Sasoe [5] we have cosec γ t 5 / 4 Hece / 0 t Copright to IJARSET wwwijarsetcom 59

8 With the help of above estimates we are i a positio to calculate asmptotic estimates of X, Y ad Z Usig 0 ad i we have t Hece π t π O t From 6, ad, we have t / O t 5 / t 6t π t So that π t t O t O t 5 / Usig the above estimates i 0, ad, we obtai / X t O 5 π Y O t / 6 Z t O t 7 π From the defiitio of, we have XZ Y t O t 9 π From 8 ad the fact that K / 0 as, we have K X / O 7δ From 8 ad 9, we have Copright to IJARSET wwwijarsetcom 50

9 KY / X O From 5 ad 9, we have 7δ ZK 7δ O 0 From 0 it is clear that Sice ZK 7δ exp 7 δ teds to zero for large, ad ZK exp Also from 8 ad, we have X t / Cosequetl, we have Sice erf X 0 I, ~ I, exp ZK X t O / / KY / X 7 7 dt O dt K exp erf X / KZ dt X 7 Thus from ad 4, we obtai 7 EN f :, O 5 4 Copright to IJARSET wwwijarsetcom 5

10 V AVERAGE NUMBER OF LEVEL CROSSING IN THE INTERVALS -,-+ε AND - ε, We show that i the rages - ε t ad - t -+ ε, the umber of zeros of compariso to those i the iterval alread cosidered We cosider the iterval - ε,, to begi with Let Fz=f ω, z = ω z K 0 Where ω,isthe radom vector [ 0 ω, ω, ω ] 0 ω 6 K is small, i Now F=f ω = ω z K 0 is a radom variable with mea K ad variace ad so has the distributio fuctio, Hece πξ πξ πξ t t f e v exp ξ v exp ξ e e Let I max ω 0 we have, iθ Let T max e 0 The dv ξ 0 dv I e 0, / 7 Copright to IJARSET wwwijarsetcom 5

11 f e iθ 0 0 I Y ω Y ω 0 T T K K e e iθ iθ K f iθ e T K e 8 Hece we have 6 Coclusio:-B cosiderig 0 w be radom polomial such that [ 0 w, w, w] is a sequece of mutuall idepedet, ormall distributed radom variables with mea zero ad variace uit ad [Ψ 0,Ψ, ] be a sequece of ormalized orthogoal Legedre polomials, defied b, where is the classical Legedre polomial The, for a costat K such that K / 0 as, we foud the mathematical expectatio of umber of real zeros of the equatio w = is asmptotic to / 0 Hece our theorem is proved REFERENCES Crammer, H Radom Variables ad probabilit Distributios d ed Cambridge Uiversit press, 96 Das, MK Real zeros of a radom sum of orthogoal polomials, roc Amer Math Soc 7, 97, 47-5 Farahmad, K O the average umber of real roots of radom algebraic equatio, Aal Of prob 4, 986, Farahmad, K The average umber of level crossigs of radom algebraic polomial, Stoch Aal Appl 6 988, Sasoe, G Orthogoal Fuctios, Itersciece, New Yor, 959 Copright to IJARSET wwwijarsetcom 5

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of