Fourier Series nd Their Applictions Rui iu My, 006 Abstrct Fourier series re of gret importnce in both theoreticl nd pplied mthemtics. For orthonorml fmilies of complex vlued functions {φ n}, Fourier Series re sums of the φ n tht cn pproximte periodic, complex vlued functions with rbitrry precision. This pper will focus on the Fourier Series of the complex exponentils. Of the mny possible methods of estimting complex vlued functions, Fourier series re especilly ttrctive becuse uniform convergence of the Fourier series (s more terms re dded) is gurnteed for continuous, bounded functions. Furthermore, the Fourier coefficients re designed to minimize the squre of the error from the ctul function. Finlly, complex exponentils re reltively simple to del with nd ubiquitous in physicl phenomen. This pper first defines generlized Fourier series, with n emphsis on the series with complex exponentils. Then, importnt properties of Fourier series re described nd proved, nd their relevnce is explined. A complete exmple is then given, nd the pper concludes by briefly mentioning some of the pplictions of Fourier series nd the generliztion of Fourier series, Fourier trnsforms. Introduction nd Bckground Informtion In the mid eighteenth century, physicl problems such s the conduction ptterns of het nd the study of vibrtions nd oscilltions led to the study of Fourier series. Of centrl interest ws the problem of how rbitrry rel vlued functions could be represented by sums of simpler functions. As we shll see lter, Fourier series is n infinite sum of trigonometric functions tht cn be used to model rel vlued, periodic functions. We shll begin by giving brief description of the trigonometric polynomils, nd especilly of their reltion to the complex exponentils. Let us define: ix + e ix cos(x) [e ], () ix e ix sin(x) [e ]. () i
Another wy to express this is tht cos(x) is the rel prt of e ix nd tht sin(x) is the imginry prt. Let us quickly show tht both functions re periodic with period. To do so, we just need to show tht e i(x+) e ix, nd the periodicity of sin(x) nd cos(x) follow by definition. e i(x+) e ix e i e ix e (i) e ix ( ) e ix We now move to the definition of trigonometric polynomil. For complex numbers { 0,,...} nd {b, b, b 3...}, nd rel x, we define trigonometric polynomil to be finite sum of the form: f (x) 0 + ( n cos(nx) + b n sin(nx)). (3) n0 ow, for n,, 3..., define c n ( n ib n ), c n ( n + ib n ). Also, define c 0 0. Then from the bove identity, we cn lso write trigonometric polynomil in equivlent form by: inx f (x) c n e. (4) inx in n inx e ow consider the function e. Clerly, both f (x) e inx nd f (x) in e (inx) e both hve periods of. Furthermore, we know tht dx inx in p+ f (x). Then, since f (x) f (x + ), f p (x) 0 if n,, 3..., nd if p+ n 0, f (x). p ow let us multiply eqution (4) bove by e imx, where m is ny integer. We obtin the following: e imx f (x) e imx inx n [(c inx )(e imx n e )] n (c n e n c n e i(n m)x ) Consider two cses. First, suppose tht m >. In this cse, for ll n, n m 0. Thus, in this cse, e i(n m)x dx 0.
ow, suppose tht m. In this cse, there is exctly one n such tht n Z, n, nd tht is n m. Since e inx dx if nd only if n 0, where m n. Thus, Since m n, e imx f (x)dx c n e i(m n)x dx, e imx f (x)dx c n, nd c n e imx f (x)dx. c m e imx f (x)dx (5) We now define trigonometric series to be of the form where the th prtil sum is n n inx c n e, (6) inx c n e. (7) Furthermore, for given function f (x), we shll define the Fourier series of f(x) s the trigonometric series with coefficients of the form given in eqution (5). Generl Fourier Series Before focusing on Fourier series with trigonometric functions, we shll give description of generl Fourier functions. We strt with the notion of orthogonl systems of functions. Let {φ, φ, φ 3...} be series of complex functions. We sy tht {φ n } is n orthogonl system of functions on [,b] if, for ll integers m n, φ m (x)φ n(x)dx 0, (8) As further note, if for ll integers m > 0, φm (x)φ m(x)dx, 3
we sy tht {φ n } is n orthonorml system of functions. We hve lredy seen tht the functions e inx, n,, 3... form n orthogonl system of functions on [, ], since ē inx e inx, nd for m n, e inx e imx 0. We now define the Fourier coefficients with respect to {φ n (x)} s follows: c n f(x)φ n(x). (9), where φ n(x) is the complex conjugte of the complex vlued function φ(x). In terms of generlized Fourier series, define the Fourier series of f with respect to {φ n (x)} to be c n φ n (x) (0) n To see how our definition for the Fourier series with respect to trigonometric functions mtches this pttern, let nd let φ(x) e inx, φ (x) e ( inx). {φ n (x)} {φ (x), φ (x), φ 3 (x), φ 4 (x)} ix {e ix, e ( ix), e, e ( ix) } 3 Some Properties of Fourier series We now present few importnt properties of Fourier series from Wlter Rudin s Principles of Mthemticl Anlysis. Theorem 8. in Rudin: Suppose tht {φ n } is n orthonorml system of functions on the intervl [, ]. Suppose tht we hve two sets of complex numbers, c n nd d n, n 0,,, 3... nd c n re the Fourier Coefficients for {φ n }, s defined in eqution (9). ow, consider two series of functions, s (f, x) c n {φ n }, () n which is the th prtil sum of the Fourier series for f, nd Then, t (f, x) d n {φ n }. () n f s n (f, x) dx f t n (f, x) dx. (3) 4
This theorem indictes tht, for some periodic function f nd some orthonorml system of functions {φ n }, the Fourier series provides the lest totl squred error pproximtion. Proof: Let {φ n (x)} be orthonorml on the intervl [, b]. Consider: n ft n f dn φ n by the definition of t n. We cn lso write the bove s: Since n n f d nφ n fd n φ n fφ n c n, we cn once gin rewrite the bove expression s: n ft n c n d n (4) ow, consider the integrl from to b of t n. Since t n t n t n nd we hve: n t n d mφ m n t n d mφ m, n n tn d m φ m d k φ k, which we my rewrite s: n n t n d m φ m d k φ k (5) Since {φ n } is n orthonorml system of functions on [, b], ccording to eqution (8), we cn rewrite the bove s: which we cn gin rewrite s: n tn d m d m, n (6) t n d m ow, consider the totl squred error between f nd t n, f t n. We first rewrite it s: 5
f tn (f t n )(f t n ) Furthermore, we know tht (f t n ) f tn, so tht f t n (f t n )(f tn ) (ff ft n ft n + t n t n ) (f ft n ft n + t n ) By equtions (4) nd (6) bove, we cn write: n n n Since f t n (f ) (c m d m) ( c m d m ) + (d m d m) (7) d m c m (d m c m )(d m c m) d m + c m (c m d m) ( c m d m ) We cn rewrite the bove eqution s: n n f t n (f ) c m + d m c m (8) From this bove eqution, we cn see tht the totl error squred is minimized when d m c m, for m,, 3.... Theorem 8. in Rudin: Assume ll the nottion used in the description of Theorem 8.. Consider the sequence of terms {c n } c, c, c 3.... The sen n ries (c m) converges bsolutely (in other words, the series c m converges). Proof: In eqution (6) bove, substitute c n for d n. We obtin the following: n s n (x)dx c m (9) The bove step will not be necessry, but it is interesting to point out tht the integrl of the bsolute vlue of ny n t h Fourier trigonometric polynomil will be less thn the integrl of the bsolute vlue of the function f. 6
ow consider eqution (8). Since we know tht tht n c m If we let n go to infinity, we see tht: c n f t n 0, it follows f(x) dx (0) f(x) () From our study of convergent series, this lso implies tht lim n c n 0 This result is firly importnt becuse it shows us tht it is the first terms of Fourier series tht re most importnt, nd tht the Fourier coefficients become rbitrrily smll. In terms of simultions, this implies tht few terms my provide very good model of function. Theorem 8.4 in Rudin: For periodic function f(x), suppose tht for some x, there is δ > 0 nd some finite, rel M such tht if δ < t < δ, then f(x + t) f(x) M t. Then, the vlue of the infinite Fourier series s n (f, x) evluted t x converges to f(x) (evluted t x) s n pproches infinity. This theorem tlks bout the pointwise convergence of Fourier series. At ll points with the property bove, the series of Fourier polynomils converges pointwise to f() t. An interesting consequence of this result is tht for some function f(x) tht is uniformly continuous on some segment (, b), the Fourier series will converge to the function f(x) in vlue for ll x in tht segment. A stronger result tht describes the uniform convergence of the Fourier series follows. Theorem 8.5 in Rudin: This is nother theorem of convergence, lthough it does not mention Fourier series explicitly. Suppose tht f(x) is continuous with period. Then, for ny ɛ > 0, there is some trigonometric polynomil such tht P (x) f(x) < ɛ for ll x. Another wy of stting this is tht given ny continuous, periodic, complexvlued function f, there is some sequence of trigonometric polynomils tht converges uniformly to f. This is bsiclly direct ppliction of the Stone Weierstruss Theorem for complex vlued functions. The metric spce K of the complex exponentils is the unit circle, which is closed nd bounded in R, nd therefore compct. The fmily of trigonometric functions is n lgebr of functions, since it is closed under ddition, multipliction, nd sclr multipliction. It clerly seprtes ll points, nd vnishes t no points. Furthermore, it is self djoint for every trigonometric polynomil f(x), there is trigonometric polynomil f (x). Thus, the fmily of trigonometric polynomils is dense in K. 7
4 An Exmple We now pply the Fourier series to few bsic exmples. Let us first consider the Fourier series for f (x), which is continuous function with period, nd whose vlue on the intervl [, ] is x. We know tht the Fourier coefficients c n re given by: Since we know tht c n f (x)e inx dx xe inx + e inx ) xe inx dx ( in n e inx c n (inx + ) n e in e in ( n (in + ) ( in + ) n ) (cos(n) isin(n))(in + ) (cos(n) + isin(n))( in + ) n ( ) incos(n) isin(n) + incos(n) isin(n) n (ncos(n) sin(n))i n nd inx c n ( inx + ) en e in e in ( in + ) (in + ) n n ( ncos(n) + sin(n))i n This lmost gives us ll the informtion we need to determine the Fourier series. But we first need to clculte c 0, since the method of integrting e 0 is different. So, c 0 xe 0 ( ) 0 8
ow we cn construct the Fourier series for f (x). For exmple, let s tke the first 9 terms (or n 0,,, 3, 4). i i i i We hve: c 0 0, c i, c i, c, c, c 3 3, c 3 3, i c 4 4, c 4 i 4, nd the 4th prtil sum of the Fourier series of f (x) is given by: s (f, x) 0 ie ix + ie ix + ( i e ix ) i e ix i e 3ix +... 3 We cn now combine terms rised to the negtive exponents of ech other, nd rewrite ech complex exponentil in terms of its sine nd cosine terms. Doing so, we obtin the formul s (f, x) 0 + sin(x) sin(x) + sin(3x) sin(4x) 3 Grphing this function, we cn indeed see the shpe of f (x) beginning to pper. 9
4 Figure : Fourier Series with First 5 Terms 3 0 3 4 0 8 6 4 0 4 6 8 0 4 Figure : Fourier Series with First 5 Terms 3 0 3 4 0 8 6 4 0 4 6 8 0 4 Figure 3: Fourier Series with First 30 Terms 3 0 3 4 0 8 6 4 0 4 6 8 0 Furthermore, if we clculte the totl squred error on the intervl [, ], we find tht: (x s (f, x)).78. Indeed, if we use the first 4 terms of the Fourier series (the 0th prtil sum), the function ppers much more similr to f (x), nd the squre error from to decreses drmticlly. Sometimes, leving the coefficients in the forms {c 0, c, c...} is not very convenient. We will now use some properties of complex exponentils to crete nicer form for the Fourier series. 0 First, since the term ssocited with c 0 in the Fourier series is c 0 e, we cn replce it with some rel constnt. Let s cll this constnt 0 c 0. For c n nd c n, n,, 3..., recll the definitions tht 0
c n ( n ib n ) c n ( n + ib n ) Then, n c n + c n b n i(c n c n ) Since we know tht c n is the complex conjugte of c n, we cn sy tht n is twice the rel prt of c n (or c n ), nd b n is negtive two times the rel coefficient of the imginry prt of c n (or c n ). These reltions let us put the Fourier series into n lternte form, given by: f (x) 0 + ( n cos(nx) + b n sin(nx)) () n0 where n nd b n re given s bove. 5 Applictions of Fourier series To recpitulte, Fourier series simplify the nlysis of periodic, rel vlued functions. Specificlly, it cn brek up periodic function into n infinite series of sine nd cosine wves. This property mkes Fourier series very useful in mny pplictions. We now give few. Consider the very common differentil eqution given by: x (t) + x (t) + b f (t) (3) This eqution describes the motion of dmped hrmonic oscilltor tht is driven by some function f (t). It cn be used to model n extensive vriety of physicl phenomen, such s driven mss on spring, n nlog circuit with cpcitor, resistor, nd inductor, or string vibrted t some frequency. There re two prts to the solution of eqution (5). The first prt is trnsient tht fdes wy (generlly) firly quickly. When the trnsient is gone, wht remins is the stedy stte solution. This is wht we will concern ourselves with. If f (t) is sinusoid, then the solution is lso sinusoid which is not very difficult to find. The problem is tht the driver is generlly not simple sinusoid, but some other periodic function. In electronics, for exmple, common driving voltge function is the squre wve s(t), periodic function (whose period we shll sy is ) such tht s(t) 0 for t < 0 nd s(t) for 0 t <. The physicl property of oscillting systems tht mkes Fourier Anlysis useful is the property of superposition in other words, suppose the driving force
f (t), long with some initil conditions, produces some stedy stte solution x (t), nd tht nother driving force, f (t) produces the stedy stte solution x (t). Then the driving force f 3 (t) f (t) + f (t) produces the stedy stte response x 3 (t) x (t) + x (t). Then, since we cn represent ny period driving function s Fourier series, nd it is simple mtter to find the stedy stte solution to sinusoidllydriven oscilltor, we cn find the response to the rbitrry driving function f (x) 0 + ( n cos(nx) + b n sin(nx)). So suppose we hd our squre wve eqution, where f (t) is the squre wve function. We could then decompose the squre wve into sinusoidl components s follows: c s(x)e inx n e inx 0 i in (e ) n c n i (e in ) n nd then just combine the c n nd c n terms s before. The result would be n infinite sum of sin nd cos terms of the form in eqution (). The stedy stte response of the system to the squre wve would then just be the sums of the stedy stte responses to the sinusoidl components of the squre wve. The bsic equtions of the Fourier series led to the development of the Fourier trnsform, which cn decompose non periodic function much like the Fourier series decomposes periodic function. Becuse this type of nlysis is very computtion intensive, different Fst Fourier Trnsform lgorithms hve been devised, which lower the order of growth of the number of opertions from order( ) to order(n log(n)). With these new techniques, Fourier series nd Trnsforms hve become n integrl prt of the toolboxes of mthemticins nd scientists. Tody, it is used for pplictions s diverse s file compression (such s the JPEG imge formt), signl processing in communictions nd stronomy, coustics, optics, nd cryptogrphy. References [] Wlter Rudin. Principles of Mthemticl Anlysis, Third Edition McGrw Hill Interntionl Editions (976). [] Eric W. Weisstein. Fourier Series. From Mthworld Wolfrm Web Resource. http://mthworld.wolfrm.com/fourierseries.html
[3] Eric W. Weisstein. Fourier Trnsforms. From Mthworld Wolfrm Web Resource. http://mthworld.wolfrm.com/fouriertrnsforms.html 3