Orthogonal Functions. Orthogonal Series Expansion. Orthonormal Functions. Page 1. Orthogonal Functions and Fourier Series. (x)dx = 0.

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Claud Quinn
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1 Orthogol Fuctios q Th ir roduct of two fuctios f d f o itrvl [, ] is th umr Orthogol Fuctios d Fourir Sris ( f, f ) f f dx. q Two fuctios f d f r sid to orthogol o itrvl [, ] if ( f, f ) f f dx. q A st of rl-vlud fuctios {φ, φ, φ, } is sid to orthogol st o itrvl [, ] if (ϕ m,ϕ ) ϕ m ϕ dx, m. q Exmls: Th st {, cos x, cos x, } o th itrvl [-π, π] is orthogol. 7 Orthoorml Fuctios q Th orm or grlizd lgth of fuctio is dfid s ϕ (ϕ,ϕ ) ϕ dx. q A st of orthogol fuctios {φ, φ, φ, } tht r ormlizd y thir orms is clld orthoorml st.!# ϕ ϕ, ϕ ϕ, ϕ $# ϕ,! %# & # q Suos {φ } is ifiit orthogol st of fuctios o itrvl [, ] d yf is fuctio dfid o this itrvl. Th, whr Orthogol Sris Exsio f q This is clld orthogol sris xsio of f. ϕ, f ϕ dx, ϕ q Exmls: Th st o th itrvl [-π, π] is orthogol. { / π,cos x / π, cosx / π,!} 8 9 Pg

2 Orthogol Fuctios with Wight Fuctio q A st of rl-vlud fuctios {φ, φ, φ, } is sid to orthogol with rsct to wight fuctio w o itrvl [, ] if wϕ m ϕ dx, m. q Suos {φ } is ifiit orthogol st of fuctios o itrvl [, ] d yf is fuctio dfid o this itrvl. Th, whr q This is clld orthogol sris xsio of f. f ϕ, f wϕ dx, ϕ ϕ wϕ dx. q Th st of trigoomtric fuctios π π 3π π π 3π,cos x,cos x,cos x,!,si x,si x,si x,! is orthogol o th itrvl [-, ]. q Th Fourir sris of fuctio f dfid o th itrvl (-, ) is giv y Fourir Sris f +! # cos π x + si π x $ & % f dx, f cos π x dx, f si π x dx Covrgc of Fourir Sris q Lt f d f icwis cotiuous o th itrvl (-, ); tht is f d f cotiuous xct t fiit umr of oits i th itrvl d hv oly fiit discotiuity t ths oits. q Th Fourir sris of f o th itrvl covrgs to f t oit of cotiuity. q At oit of discotiuity, th Fourir sris covrgs to th vrg [ f (x + )+f (x - )]/ whr f (x + ) d f (x - ) dot th limit of f t x from th right d from th lft, rsctivly. Clss Exrcis q Fid d lot Fourir sris of #%, π / < x < f $ &% cos x, x < π / π π / π / cos x dx π cos x cosx dx ( )+ π π (4 ) π / 4 cos xsix dx π π (4 ) f π + # ( ) + π (4 ) cosx + 4 % $ π (4 ) six & ( 3 Pg

3 Ev d Odd Fuctios q A fuctio f is sid to v if f (-x) f d odd if f (-x) - f. q Th roduct of two v fuctios is v. q Th roduct of two odd fuctios is v. q Th roduct of v fuctio d odd fuctio is odd. q Th sum (diffrc) of two v fuctios is v. q Th sum (diffrc) of two odd fuctios is odd. q If f is v, th f dx f dx. q If f is odd, th f dx. Fourir Cosi d Si Sris q Th Fourir sris of v fuctio f dfid o th itrvl (-, ) is cosi sris giv y f + cos π x f dx, f cos π x dx q Th Fourir sris of odd fuctio f dfid o th itrvl (-, ) is si sris giv y f si π x f si π x dx 4 5 q Eulr s Formul: For rl umr x ix q Solvig for cos x d si x givs q Sustitutig ito Fourir sris of fuctio f dfid o itrvl (-, ) rsults i comlx Fourir sris of this fuctio. Comlx Fourir Sris cos x + isi x cos x ix + ix f ix si x iπ x/ cos x isi x ix i ix f iπ x/ dx,,±,±,! 6 Fourir Sris d Frqucy Sctrum q Fourir sris of fuctio o th itrvl (-, ) dfis riodic fuctio with th fudmtl riod of T. q If w dfi ωπ / T s th fudmtl gulr frqucy, th Fourir sris com f ( x) + q Th lot of oits (ω, ) is clld frqucy sctrum of f. cos ω x + si ωx d f ( x) c iωx 7 Pg 3

4 Clss Exrcis q Fid comlx Fourir sris d frqucy sctrum of $ f # %$ ( ), πi c f ( ) πi iπ x/, < x <, < x < Sturm-Liouvill Prolm q Lt, q, r, d r rl-vlud fuctios cotiuous o itrvl [, ], d lt r > d > for vry x i th itrvl. Th, d [ r( x) y ] + ( q( x) + λ( x) ) y dx A y( ) + B y( ) A y( ) + B is sid to rgulr Sturm-Liouvill Prolm. q Exml: Lgdrs Equtio : ( x y( ) ) y xy + ( + ) y 8 9 Prortis of Rgulr Sturm-Liouvill Prolm q Thr xist ifiit umr of rl igvlus tht c rrgd i scdig ordr λ < λ < λ 3 < < λ such tht λ s. q For ch igvlu thr is oly o igfuctio. q Eigfuctios corrsodig to diffrt igvlus r lirly iddt. q Th st of igfuctios corrsodig to th st of igvlus is orthogol with rsct to th wight fuctio o th itrvl [, ]. Covrsio to Slf-Adjoit Form q Evry scod-ordr diffrtil qutio y+ y+ c + λd c covrtd to th so clld slf-djoit form y dividig th origil qutio y d multilyig y q Exml: Prmtric Bssl Equtio x y+ xy+ (α x v d )y dx [xy]+ # α & % x v (y $ x ( ) y d dx ry [ ] + q+ λ ( ) y (/)dx Pg 4

5 x Bssl Fuctios r Orthogol q Th rmtric Bssl qutio d y + xy + ( α x ) y [ xy] + α x y dx x hs two solutios J (αx) d Y (αx) ut oly J (αx) is oudd t x. q Th st [J x)] is orthogol with rsct to th wight fuctio x o itrvl [, ]; xj x)j (α j x)dx, α i α j. q α i r giv y oudry coditio t x ; A J ( α) + BαJ ( α). Diffrtil Rcurrc Rltios d! dx x J # $ x J d! dx x J # $ x J + or xj J xj + xj xj J 3 q Th orthogol sris xsio of fuctio f dfid o th itrvl [, ] i trms of Bssl fuctios, whr Fourir-Bssl Sris f c i J x), c i i xj x) f dx, J x) Lgdr s Fuctios r Orthogol q Th Lgdr olyomils, which r th solutios of th Lgdr s qutio ( x ) y xy + ( + ) y, r orthogol with rsct to th wight fuctio o th itrvl [-, ]; P m P dx, m. d th squr orm of th fuctio J x) is dfid y J x) xj x)dx. 4 5 Pg 5

6 q Th orthogol sris xsio of fuctio f dfid o th itrvl [-, ] i trms of Lgdr fuctios, whr Fourir-Lgdr Sris f P, P f dx, P d th squr orm of th fuctio P is dfid y P + 6 Pg 6

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series