ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces. Every vector in R n cn be represente s sum of the form 3.1 v = v i e i i=1 where e i is the unit vector long the i th coorinte xis. However if ff i g is ny set of n linerly inepenent vectors then we lso hve unique representtion of v s 3.2 v = c i f i : i=1 It shoul be note however tht the vectors f i nee not be orthogonl nor nee they hve unit length for the expnsion 3.2 to work. If however 3.3 f i f j = ij ; tht is if the f i re orthonorml then the coecients c i cn esily be compute. For if we tke the ot prouct of v with bsis vector f i we get tht is to sy f i v = f i @ X n c j f j 1 A = n X c j f i f j = c j ij = c i ; 3.4 c i = f i v : 2. Bses of Orthogonl Functions The relevnce of these remrks now comes from the observtion tht the set CëR; Rë of continous rel-vlue functions on the rel line is lso vector spce; for the opertions of ition n sclr multipliction of functions re well-ene: ëf + gëx = fx + gx ëc fëx = cfx 8 c 2 R: Note however tht CëR; Rë is n innite imensionl vector spce. Inee the Tylor expnsion 2.16 of cn be thought of s n expnsion of with respect to bsis of monomil functions ff mn x; t =x m t n g. Unfortuntely this bsis is not orthonorml t lest not with respect to ny obvious inner prouct. Fourier series re better exmples since they constitute expnsions of functions with respect to n orthonorml bsis of CëR;Rë. 1
2. BASES OF ORTHOGONA FUNCTIONS 11 Theorem 3.1. Fourier Theorem If f is function such tht fx n f x tht re piecewise continuous on the intervl ë;ë R then the series 1X nx 1X nx 2 + n cos + b n sin where the coecients n b n re etermine by converges pointwise to fx for ll x 2 ë;ë. R nx cos n = 2 b n = 2 R sin 2x fx fx In view of the formuls 2 mx sin 2 mx sin 2 mx cos sin cos cos nx nx nx x = m;n x = x = m;n the Fourier expnsion of f mounts to expnsion of f with respect to the bsis sin mx which is orthonorml with respect to the inner prouct f g = 2 fxgx : ; cos nx Sturm-iouville theory is generliztion of Fourier theory. It provies mens of constructing other sets of orthonoml bses for spces of functions. Theorem 3.2. Sturm-iouville Theorem Consier bounry vlue problem of the form 3.5 px y + qx + rx y = c1y +c2y = 1yb +2y b = where px n rx re smooth positive functions on the intervl ; b. Then i For ll but iscrete set S of choices of there re no solutions to 3.5. There exists miniml 2 S. n one cn rrnge the set S of missible so tht with One then hs S = f;1;2;::: ;g é1 é2 é lim n!1 n =+1 : ii To ech eigenvlue n 2 S there correspons exctly one solution n x of 3.5. iii If n ; m 2 S n n x m x re the corresponing solutions then n 6= m n x m xrx = :
2. BASES OF ORTHOGONA FUNCTIONS 12 iv The set of functions 8 é é: m = m x hr b m x 2 rxs i 1=2 m =; 1; 2; 9 é= é; form complete orthonorml bsis for spce of piecewise continuous functions on the intervl ë; bë. Tht is to sy every piecewise continous function f :ë; bë! R cn be expne s 3.6 1X fx n n t where the coecients n re etermine by 3.7 3.8 The series 3.6 converges in the sense tht lim N!1 n = fx n xrx : fx NX 2 n n x! = : Remrks: i Existence of Eigenvlues n Eigenfunctions. Sttement i is me plusible by consiering the simple exmple y + 2 y = y = y = 9 = ; = n nx yx = A sin n inee in pplictions before one cn mke use of the expnsion 3.6 one hs to rst n the eigenvlues i n so prt i my be regre s prove constructively. However n bstrct proof lso exists. ii Uniqueness of Eigenfunctions. Sttement ii follows from the existence n uniqueness theorem for secon orer liner ODE's. iii Orthogonlity Property of Eigenfunctions. et 1 n 2 be solutions of 3.5 for = 1 n = 2 respectively. Assume 1 6= 2. Then px 1 +qx +1 rx 1 = px 2 +qx +2 rx 2 = Multiplying the rst eqution by 2 n the secon by 1 n subtrcting the two equtions yiels 1 2 rx 1 2 = 2 px 1 1 px 2
3. EXAMPES 13 Integrting both sies of the expression bove between x = n x = b yiels 1 2 1 x 2 xrx = 2 x px 1 x 1 x px 2 x = 1 x 2 xpx b 1 x 2 xpx b + 1 x 2 xpx 1 x 2 xpx = rb 1 b 2 b 1 b 2 b r 1 2 1 2 Now in orer for the bounry conitions t x = c1 1 +c2 1 = c1 2 +c2 2 = to hve solutions with c1;c2 not both zero we must hve 1 2 1 2 = : This cn be seen s follows: Recll from liner lgebr tht if M is n n mtrix n v is n-imensionl vector then Mv = hs non-trivil solutions if n only if et M =. The sttement bove then follows by consiering 1 M = 2 1 2 c1 ; v = c2 Similrly in orer for the bounry conitions t x = b to be stise for c3;c4 not both zero we must hve 1 b 2 b 1 b 2 b = : : Thus we hve So if 1 6= 2. 1 2 1 x 2 xrx = 1 x 2 xrx = iv Completeness of Eigenfunctions The hr thing to unerstn is the remrkble completeness property expresse in sttement iv. The proof of this sttement is not terribly icult - however it oes require moerte igression into the Clculus of Vritions. At the en of the course time-permitting we will evelop the Clculus of Vritions n then prove iv s smple ppliction. 3. Exmples Exmple 3.3. Fourier Sine Series: y + y = y = y =
3. EXAMPES 14 This is Sturm-iouville type problem with px = rx = 1qx =. The generl solution of the ODE is given by p p yx =Asin x + B cos x however such solutions will stisfy the bounry conitions y = y = if n only if B = n p = n ; n =1; 2; 3;::: : The Sturm-iouville inner prouct is n the functions f; g = y n = fxgx r 2 nx sin constitute complete orthonorml bsis for the set of piecewise continuous functions on the intervl ;. Exmple 3.4. Fourier Cosine Series: y + y = y = y = This is Sturm-iouville type problem with px = rx = 1qx =. The generl solution of the ODE is given by p p yx =Asin x + B cos x however such solutions will stisfy the bounry conitions y = y = if n only if A = n p = 2n ; n =1; 2; 3;::: : The Sturm-iouville inner prouct is n the functions f; g = y n = fxgx r 2 2nx cos constitute complete orthonorml bsis for the set of piecewise continuous functions on the intervl ;. Exmple 3.5. Bessel Functions Exmple 3.6. egenre Functions 3.9 Exmple 3.7. Hermite Functions 3.1 x J n; n2 2 x J n; + x + J n; = 1 x 2 l e x2 H +1+ll + 1 l = +2e x2 H = In ech of the lst three exmples just s in the cse of the rst two exmples there exist ierent sets of orthogonl functions epening on the bounry contions impose.