The structure of Fourier series

Transcription

1 The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial fuctio with egative base, viz 1) Keywors: Fourier trasform, expoet, egative base We ispect the epeece, say, o time of a boue quatity f expaig it ito the sum of perioic processes w First of all, cosier a iscrete process f j, reere by the vector comprise of values, f = f 0,, f j,, f,, f ), a, accorigly, expa it ito the sum of vectors w = w 0,, w j,, w,, w ): f = c w = c 0 w 00 w j0 w 0 w c j w 0j w jj w j w j + + c We see for the coefficiets c of the expasio 1) The simplest iscrete perioic process is escribe by the vector w 0 w j w w + + c w 0 w j w w 1) w = 1, q, q 2,, q j,, q,, q ), where q = 1 2) Process 2) is the oscillatio w j = 1) j betwee 1 a 1 epeig o iteger argumet j a has the perio 2 We will geeralize this cofiguratio from q = 1 to The process w, q = 1) 2 3) w = 1, q, q 2,, q j,, q,, q ), 4) will have perio /: startig from w 0 = 1 through j = / poits there will be agai w j = q j = 1, a at the itermeiate value j = /2) the fuctio is w j = q j = 1 At = 0 the perio of the oscillatio is ifiite, ie the quatity is costat At = /2 the process is ietical to 2) Whe > /2 the perio of the oscillatio will be less tha 2 I the whole, the frequecy of the processes varies i the rage 0 / < 1 The set of vectors w is orthogoal i the sese that w j w = w 1 mj w m = q m j q m = 1) 2 j)m = δ j 5) where δ j is the Kroecer elta Equality 5) follows from the formula of geometric progressio with the eomiator p = 1) 2 j)/ 1 + p + p p = 1 p )/1 p) 6) Electroic aress: aether@yaexru

2 The property of orthogoality of vectors w is coveiet by that multiplyig the expasio 1) by vector w, there ca be immeiately, maig use of 5), fou coefficiet c : w f = w mf m = c 7) 2 I the result we obtai f j = c = 1 2 c 1) j, 8) f m 1) 2 m 9) The fractioal power of 1, as i 8) a 9), ca be reuce to 1 a reere i the expoetial form with a positive base Theorem: Proof for efiitio of the Euler s umber e see Appeix) Iee, o the oe sie we have: O the other, by the trigoometry: e 1ϕ = cos ϕ + 1 si ϕ 10) e 1ϕ1 e 1ϕ2 = e 1ϕ1 + ϕ 2 ) cos ϕ si ϕ 1 )cos ϕ si ϕ 2 ) = cosϕ 1 + ϕ 2 ) + 1 siϕ 1 + ϕ 2 ) Besies, ifferetiatig the left-ha sie of 10) we obtai While the ifferetial of the right-ha sie 10) is ϕ e 1ϕ = 1e 1ϕ ϕ cos ϕ + 1 si ϕ) = si ϕ + 1 cos ϕ = 1cos ϕ + 1 si ϕ) This is sufficiet i orer to substatiate equality 10) Cosier the chage of 1) x i epeece o x: 1) 0 = 1, 1) 1/2 = 1, 1) 1 = 1, 1) 3/2 = 1, 1) 2 = 1 Accorig to 10), we have exp 1π) = 1 a, cosequetly, exp 1πx) chages with x as e 1π 0 = 1, e 1π 1/2 = 1, e 1π 1 = 1, e 1π 3/2 = 1, e 1π 2 = 1 Thus we have show that 1) x = e 1πx 11) Now we have the coveiet form 11), 10) which visually emostrates the oscillatio as rotatio of a uit vector i cooriates 1, 1) Fig1 may illustrate the istributio o the plae i these cooriates of compoets of a vector 4) with the eomiator 3)

3 3 Figure 1: The proucts exp[i j)m/] of the compoets of orthogoal vectors i relatio 5) for the case of the basis cosistig of = 5 vectors are show o the complex plae: m-th compoet of j-th a -th vectors, j = 2, m = 0, 1, 2, 3, 4 Usig 11) i 8) a 9), we fi the staar form of the Fourier expasio f j = c = 1 c e i j, 12) where the well-ow esigatio for the imagiary uit 1 = i is assume Next we procee to the cotiuous presetatio supposig Usig 14) i 12), 13): ft) = f m e i m 13) t = j t, T = t = cost, t 0 14) c e i t j t c = 1 f m e t m t t 1 t T c e i T t, 15) o ft)e i T t t 16) I 15) a 16) T is the time uratio of the process, a T/ perio of the -th harmoics Replacig i 15) a 16) ft) by ft + t 0 ) we obtai formulae for expasio of the fuctio at ay fiite iterval t 0, t 0 + T ) Notice that we may exte the geometric progressio 6) at the same legth ito the regio of egative powers: p + p p p + p p = p p )/1 p) 17) I this evet, for the extee vectors w there hols the orthogoality with q = 1) /, so that we will have istea of 5) w j w = m= q m j q m = m= 1) j)m = 2δ j 18)

4 The symmetrical expasio has a more regular character: the perios of the iscrete processes i questio ever become less tha 2 At a give the perio equals to 2/, ie two times more log tha it is at the same i the oe-sie istributio 4) At = the perio equals to 2, the with the icrease of it grows up to ifiity at = 0, a further rops graually to almost 2 at = 1 So that the frequecy varies i the rage 1/2 /2) < 1/2 I accor with 18), we recast relatios 15) a 16) puttig 2 i place of : ft) = = c = 1 2T T c e iπ T t, 19) ft)e iπ T t t 20) where the fuctio is tae at a fiite iterval T, +T ) If ft) is efie o the iterval t 1, t 2 ), the we have i 19), 20) T = t 2 t 1 )/2, a ft) shoul be substitute by ft + t 1 + t 2 )/2) Notice that if the fuctio f is real, the 20) etails c = c, where is the sig of complex cojugatio We will represet the expasio of the real fuctio i the real form From 19), usig Euler formula 10): ft) = c 0 + c e iπ T t + c e iπ T t [ = c 0 + c + c ) cos π T t + ic c ) si π ] T t 21) =1 Coefficiets c 0, c + c, ic c ) of this expasio are real a ca be etermie from 20) as c + c = 1 T ic c ) = where c 0 = c 0 Thus, at a fiite iterval a fuctio ca be expae ito the Fourier series Let the time iterval be ifiite: T We will reefie variables: Usig 24) i 19) yiels ft) = 1 T T T π T = ω, T c π = =1 cω) π T eiωt sice varyig by oe ω chages by π/t, ie δω = π/t Usig 24) i 20): cω) = 1 ft) cos π tt, 22) T ft) si π tt 23) T cω) 24) cω)e iωt ω 25) ft)e iωt t 26) So, we eal with the Fourier itegral o the etire umber axis There ca be euce a formula for the cocise reerig a rememberig of the Fourier expasio First, refashio 25) as ft) = ω cω )e iω t 27) 4

5 5 Substitutig 27) ito 26): From 28) we have cω) = ω cω ) 1 te iω ω)t 28) δω ω) = 1 Similarly to 29), there ca be writte δ-fuctio for t: δt t) = 1 te iω ω)t 29) ωe iωt t) 30) Usig the represetatio 30), we may easily obtai formula for the Fourier trasform ft) = t ft )δt t) = ω 1 t ft )e iωt e iωt 31) Compare 31) with 25) a 26) Let ft) = c e iω t 32) ie we have a set of harmoic oscillators Substitutig 32) i 26): cω) = 1 c e iω ω)t t = c δω ω ) 33) where the efiitio 29) is use I reality, 32) is blurre, a the iscrete spectrum 33) egraes ito the sum of Gauss compoets cω) = c exp [ ω ω ) 2 ] σ 2 2σ 2 34) that is show i Fig2 Figure 2: A realistic spectrum of the composite sigal

6 6 Appeix A: THE FIRST REMARKABLE LIMIT Biomial 1 + 1/ raise to the power at is boue by excess a eficiecy i the followig way: < ) = ) 1 1) 2) ! 2 3! ! + 1 3! + < = = 3 Deotig lim = e A1) ) we are seeig for Supposig x = 1/ gives e x + x e x x ex = lim = e x lim x 0 x x 0 e x 1 lim = lim x 0 x e x 1 x ) 1 1 = 1 Hece: Relatioship A1) ca be geeralize We have S 1 = S 2 = x ex = e x 1 + m ) m 1) m ) 2 = m + m2 2! 2! ) m = 1 + m 1 ) 2 mm 1) S 1 2! + m3 3! +, Therefore: [ e m = lim 1 + ) 1 ] m = lim 1 + m )