Orthogonal polynomials

Transcript

1 Orthogol polyomils We strt with Defiitio. A sequece of polyomils {p x} with degree[p x] for ech is clled orthogol with respect to the weight fuctio wx o the itervl, b with < b if { b, m wxp m xp x dx h δ m with δ m :, m. The weight fuctio wx should be cotiuous d positive o, b such tht the momets exist. The the itegrl µ : f, g : wx x dx,,,,... wxfxgx dx deotes ier product of the polyomils f d g. The itervl, b is clled the itervl of orthogolity. This itervl eeds ot to be fiite. If h for ech {,,,...} the sequece of polyomils is clled orthoorml, d if p x k x + lower order terms with k for ech {,,,...} the polyomils re clled moic. Exmple. As exmple we tke wx d, b,. Usig the Grm-Schmidt process the orthogol polyomils c be costructed s follows. Strt with the sequece {, x, x,...}. Choose p x. The we hve sice Further we hve sice d p x x, x, p x p x, p x p x, x x, x, dx d x, p x x x, p x p x, p x p x x, p x p x, p x p x x x,, x, x x, x x, x, x x x dx 3, x, x x dx x dx. x 3 x x x + 6, x x dx 4 6 x x + dx The polyomils p x, p x x d p x x x + 6 re the first three moic orthogol polyomils o the itervl, with respect to the weight fuctio wx.

2 Repetig this process we obti d so o. p 3 x x 3 3 x x, p 4x x 4 x x 7 x + 7, p 5 x x 5 5 x4 + 9 x3 5 6 x x 5, The orthoorml polyomils would be q x p x/ h, etceter. q x p x/ h 3x /, q x p x h 6 5 q 3 x p 3x 7 x 3 3 h3 x x, x x +, 6 All sequeces of orthogol polyomils stisfy three term recurrece reltio: Theorem. A sequece of orthogol polyomils {p x} stisfies where p + x A x + B p x + C p x,,, 3,..., A k +,,,,... d C A h,,, 3,.... k A h Proof. Sice degree [p x] for ech {,,,...} the sequece of orthogol polyomils {p x} is lierly idepedet. Let A k + /k. The p + x A xp x is polyomil of degree. Hece p + x A xp x c k p k x. The orthogolity property ow gives p + x A xp x, p k x This implies c m p m x, p k x c k p k x, p k x c k h k. m h k c k p + x A xp x, p k x p + x, p k x A xp x, p k x A p x, xp k x. For k < we hve degree [xp k x] < which implies tht p x, xp k x. Hece: c k for k <. This proves tht the polyomils stisfy the three term recurrece reltio p + x A xp x c p x + c p x,,, 3,.... Further we hve k h c A p x, xp x A h c A h. k A h This proves the theorem.

3 Note tht the three term recurrece reltio for sequece of moic k orthogol polyomils {p x} hs the form p + x xp x + B p x + C p x with C h h,,, 3,.... A cosequece of the three term recurrece reltio is Theorem. A sequece of orthogol polyomils {p x} stisfies p k xp k y k p+xp y p + yp x,,,,... h k h k + x y d {p k x} h k k h k + p +xp x p + xp x,,,,.... Formul is clled the Christoffel-Drboux formul d its cofluet form. Proof. The three term recurrece reltio implies tht d Subtrctio gives p + xp y A x + B p xp y + C p xp y p + yp x A y + B p yp x + C p yp x. p + xp y p + yp x A x yp xp y + C [p xp y p yp x]. This leds to the telescopig sum p k xp k y x y h k This implies tht k x y k p k+ xp k y p k+ yp k x A k h k p k xp k y p k yp k x A k h k k p +xp y p + yp x k x y. A h h p k xp k y h k p +xp y p + yp x A h k h k + p + xp y p + yp x, which proves. The cofluet form the follows by tkig the limit y x: p + xp y p + yp x lim y x x y lim y x p x p + x p + y p + x p x p y x y p xp +x p + xp x. 3

4 Zeros Theorem 3. If {p x} is sequece of orthogol polyomils o the itervl, b with respect to the weight fuctio wx, the the polyomil p x hs exctly rel simple zeros i the itervl, b. Proof. Sice degree[p x] the polyomil hs t most rel zeros. Suppose tht p x hs m distict rel zeros x, x,..., x m i, b of odd order or multiplicity. The the polyomil p xx x x x x x m does ot chge sig o the itervl, b. This implies tht wxp xx x x x x x m dx. By orthogolity this itegrl equls zero if m <. Hece: m, which implies tht p x hs distict rel zeros of odd order i, b. This proves tht ll zeros re distict d simple hve order or multiplicity equl to oe. Theorem 4. If {p x} is sequece of orthogol polyomils o the itervl, b with respect to the weight fuctio wx, the the zeros of p x d p + x seprte ech other. Proof. This follows from the cofluet form of the Christoffel-Drboux formul. Note tht This implies tht Hece h wx {p x} dx >,,,,.... k h k + p +xp x p + xp x {p k x} >. h k k k + p +xp x p + xp x >. Now suppose tht x,k d x,k+ re two cosecutive zeros of p x with x,k < x,k+. Sice ll zeros of p x re rel d simple p x,k d p x,k+ should hve opposite sigs. Hece we hve p x,k p x,k+ d p x,k p x,k+ <. This implies tht p + x,k p + x,k+ should be egtive too. The the cotiuity of p + x implies tht there should be t lest oe zero of p + x betwee x,k d x,k+. However, this holds for ech two cosecutive zeros of p x. This proves the theorem. Moreover, if {x,k } k d {x +,k} + k deote the cosecutive zeros of p x d p + x respectively, the we hve < x +, < x, < x +, < x, < < x +, < x, < x +,+ < b. 4

5 Guss qudrture If f is cotiuous fuctio o, b d x < x < < x re distict poits i, b, the there exists exctly oe polyomil P with degree such tht P x i fx i for ll i,,...,. This polyomil P c esily be foud by usig Lgrge iterpoltio s follows. Defie px x x x x x x d cosider px P x fx i x x i p x i fx i x x x x i x x i+ x x x i x x i x i x i x i+ x i x. i i Let {p x} be sequece of orthogol polyomils o the itervl, b with respect to the weight fuctio wx. The for x < x < < x we tke the distict rel zeros of the polyomil p x. If f is polyomil of degree, the fx P x is polyomil of degree with t lest the zeros x < x < < x. Now we defie fx P x + rxp x, where rx is polyomil of degree. This c lso be writte s This implies tht fx wxfx dx i p x fx i x x i p x i + rxp x. i wxp x fx i x x i p x i dx + wxrxp x dx. Sice degree[rx] the orthogolity property implies tht the ltter itegrl equls zero. This implies tht wxfx dx λ,i fx i with λ,i : i wxp x x x i p dx, i,,...,. x i This is the Guss qudrture formul. This gives the vlue of the itegrl i the cse tht f is polyomil of degree if the vlue of fx i is kow for the zeros x < x < < x of the polyomil p x. If f is ot polyomil of degree this leds to pproximtio of the itegrl: wxfx dx λ,i fx i with λ,i : i wxp x x x i p dx, i,,...,. x i The coefficiets {λ,i } i re clled Christoffel umbers. Note tht these umbers do ot deped o the fuctio f. These Christoffel umbers re ll positive. This c be show s follows. We hve λ,i wxl,i x dx with l,i x : 5 p x x x i p, i,,...,. x i

6 The l,i x l,ix is polyomil of degree which vishes t the zeros {x,k } k of p x. Hece l,ix l,i x p xqx for some polyomil q of degree. This implies tht wx l,ix l,i x dx by orthogolity. Hece we hve λ,i Now we c lso prove wxl,i x dx wxp xqx dx wx {l,i x} dx >. Theorem 5. Let {p x} be sequece of orthogol polyomils o the itervl, b with respect to the weight fuctio wx d let m <. The we hve: betwee y two zeros of p m x there is t lest oe zero of p x. Proof. Suppose tht x m,k d x m,k+ re two cosecutive zeros of p m x d tht there is o zero of p x i x m,k, x m,k+. The cosider the polyomil The we hve gx Now the Guss qudrture formul gives p m x x x m,k x x m,k+. gxp m x for x / x m,k, x m,k+. wxgxp m x dx λ,i gx,i p m x,i, where {x,i } i re the zeros of p x. Sice there re o zeros of p x i x m,k, x m,k+ we coclude tht gx,i p m x,i for ll i,,...,. Further we hve λ,i > for ll i,,..., which implies tht the sum t the right-hd side cot vish. However, the itegrl t the left-hd side is zero by orthogolity. This cotrdictio proves tht there should be t lest oe zero of p x betwee the two cosecutive zeros of p m x. i y x The polyomils q x, q 3 x d q 4 x 6

7 Clssicl orthogol polyomils The clssicl orthogol polyomils re med fter Hermite, Lguerre d Jcobi: me p x wx, b Hermite H x e x, Lguerre Jcobi x e x x α, P α,β x x α + x β, Legedre P x, The Hermite polyomils re orthogol o the itervl, with respect to the orml distributio wx e x, the Lguerre polyomils re orthogol o the itervl, with respect to the gmm distributio wx e x x α d the Jcobi polyomils re orthogol o the itervl, with respect to the bet distributio wx x α + x β. The Legedre polyomils form specil cse α β of the Jcobi polyomils. These clssicl orthogol polyomils stisfy orthogolity reltio, three term recurrece reltio, secod order lier differetil equtio d so-clled Rodrigues formul. Moreover, for ech fmily of clssicl orthogol polyomils we hve geertig fuctio. I the sequel we will ofte use the Kroecker delt which is defied by {, m δ m :, m for m, {,,,...} d the ottio D d dx for the differetitio opertor. The we hve Leibiz rule D [fxgx] 3 D k fxd k gx,,,,... 4 k which is geerliztio of the product rule. The proof is by mthemticl iductio d by use of Pscl s trigle idetity + +, k,,...,. k k k 7

8 Hermite The Hermite polyomils re orthogol o the itervl, with respect to the orml distributio wx e x. They c be defied by mes of their Rodrigues formul: H x wx D wx e x D e x,,,,..., 5 where the differetitio opertor D is defied by 3. Sice D + D D, we obti D + wx D [D wx] D [wxh x] [ w xh x + wxh x ] which implies tht + wx [ xh x H x ],,,,..., H + x xh x H x,,,, The defiitio 5 implies tht H x. The 6 implies by iductio tht H x is polyomil of degree. Further we hve tht H x is eve d H + x is odd d tht the ledig coefficiet of the polyomil H x equls k. The Hermite polyomils stisfy the orthogolity reltio π e x H m xh x dx δ m, m, {,,,...}. 7 To prove this we use the defiitio 5 to obti e x H m xh x dx H m xd e x dx. Now we use itegrtio by prts times to coclude tht the itegrl vishes for m <. For m we hve usig itegrtio by prts e x H xh x dx H xd e x dx k This proves the orthogolity reltio 7. e x dx π. I order to fid the three term recurrece reltio we strt with The we hve by usig Leibiz rule 4 which implies tht wx e x w x xwx. D H x e x dx D + wx D w x D [ xwx] xd wx D wx, H + x xh x H x,,, 3,

9 Combiig 6 d 8 we fid tht Differetitio of 6 gives Now we use 9 to coclude tht H x H x,,, 3, H +x xh x + H x H x,,,, H x xh x + H x H x,,,,..., which implies tht H x stisfies the secod order lier differetil equtio y x xy x + yx, {,,,...}. Filly we will prove the geertig fuctio We strt with The Tylor series for ft is e xt t H x ft e x t e x e xt t. ft with, by usig the substitutio x t u, [ ] [ d f d dt e x t t for,,,.... Hece we hve e x e xt t e x t ft This proves the geertig fuctio. f du e u ] t. t ux f D e x e x H x t e x H x t. 9

10 Lguerre The Lguerre polyomils re orthogol o the itervl, with respect to the gmm distributio wx e x x α. They c be defied by mes of their Rodrigues formul: x By usig Leibiz rule 4 we hve D [ e x x +α] k k Hece we hve where + α k This proves tht k + α k wx D [wx x ] ex x α D [ e x x +α],,,,.... D k e x D k x +α k e x + α + α α + k + x α+k e x x α k k x we lso hve for,,,... Note tht we hve Γ + α + Γk + α + xk. + α x k k,,,,..., k k! Γ + α + k! Γk + α + k + α + k, k,,,...,. k! x is polyomil of degree. Sice k α + α + k! α + k x α + k x k α + k k! + α k α + k, k,,,..., + α α +,,,,... d tht the ledig coefficiet of the polyomil x equls Further we hve d dx Lα x d dx k,,,,.... F α + ; x. 3 + α x k k + α x k k! k k k k! k + α x k k k k! Lα+ x,,, 3,.... 4

11 Now we c prove the orthogolity reltio e x x α m x Γ + α + x dx δ m, α > 5 for m, {,,,...}. First of ll, the itegrl coverges if the momets µ e x x +α dx exists for ll {,,,...}. This leds to the coditio α >. Note tht µ Γ+α+. Now we use to obti e x x α m x x dx We pply itegrtio by prts to obti m xd [ e x x +α] dx which equls zero for m <. For m we fid D xe x x +α dx k This proves the orthogolity reltio 5. m xd [ e x x +α] dx. D m xe x x +α dx e x x +α dx Γ + α +. The Lguerre polyomils c lso be defied by their geertig fuctio t α exp xt t xt. 6 The proof is strightforwrd. Strt with 3 d chge the order of summtio to obti xt α + t k x k α + k k! α + +k k x k t +k α + k k! k xt k xt k t α k t α k! α + k x k t α + k k! k! k! k! α + k + xt k. t t This proves 6. If we defie F x, t : t α exp xt, t the we esily obti x F x, t t t α exp xt t t F x, t + tf x, t x

12 d F x, t t { } α + t α + t α x t xt t exp xt t { α + x } t α exp xt, t t which implies tht The first reltio leds to or equivletly Hece we hve t F x, t + [x α + t] F x, t. t d t dx Lα d dx Lα xt d dx Lα + x d or equivletly, by usig 4, d dx Lα The secod reltio leds to or equivletly t dx Lα xt + t d dx Lα xt + + xt xt +. x + x,,,,... 7 x x L α+ x,,,,.... xt + x xt + [x α + t] xt + xt α + xt + xt + α + xt xt +. Equtig the coefficiets of equl powers of t we obti the three term recurrece reltio + + x + x α Lα x + + α x,,, 3, Note tht this c lso be writte s [ x x x Lα x ] [ + α ] x x,,, 3,....

13 Now we differetite d use 7 to obti x d dx Lα This implies tht x + x + x + + α x,,, 3,.... x d dx Lα x x + α x,,, 3, Now we differetite 9 d use 7 d 9 to fid x d dx Lα x + d dx Lα x d dx Lα x + α d dx Lα x [ d + α dx Lα x d ] dx Lα x α d + α x α d x d dx Lα dx Lα x x α d x dx Lα x. dx Lα This proves tht the polyomil x stisfies the secod order lier differetil equtio xy x + α + xy x + yx, {,,,...}. x Filly, we use the geertig fuctio 6 to prove the dditio formul L α+β+ x + y The geertig fuctio 6 implies tht L α+β+ x + yt t α β x + yt exp This proves. t α exp xt k xtk k xlβ k y,,,,.... m t t β exp t L β m yt m yt t k xlβ k y t. 3

14 Jcobi The Jcobi polyomils re orthogol o the itervl, with respect to the bet distributio wx x α + x β. They c be defied by mes of their Rodrigues formul: P α,β x wx D [ wx x ] x α + x β D [ x +α + x +β] for,,,.... By usig Leibiz rule 4 we hve [ D x +α + x +β] k This implies tht P α,β x D k x +α D k + x +β k + α + α + α k + x +α k k + β + β β + k + + x β+k + α + β k x +α k + x β+k,,,,.... k k + α + β k x k + x k,,,,.... k k This shows tht P α,β x is polyomil of degree. Note tht we hve the symmetry d P α,β P α,β x P β,α x,,,, α d + β P α,β,,,,.... I order to fid hypergeometric represettio we write for x x + α + β x + k P α,β x,,,,.... k x Now we hve for x x + k + k x x i ki i k i k i, k,,,.... i x Now we obti by chgig the order of summtios for x d,,,... x + α + β k i P α,β x k k i x x i + α + β i + k i k 4 i + k i i. x

15 Now we reverse the order i the first sum to fid for x d,,,... x + α + β i + k P α,β x x i i i + α i + k i + k i k i + β i + k x i k i Γ + α + i + k! Γi k + α + Γ + β + i k! Γ i + k + β + Γ + α + Γ + β + i i k i α k i + β + k k!. i i + k! i! k! i Γi + α + i! Γ i + β + i x i x Sice i {,,,..., } we hve by usig the Chu-Vdermode summtio formul i k i α k i, i α F i + β + k k! i + β + ; + α + β + i. i + β + i Hece we hve by usig Γ i + β + i + β + i Γ + β + P α,β Γ + α + i + α + β + i x i x Γi + α + i! i Γ + α + i + α + β + i x i,,,,.... Γα + α + i i! i This proves the hypergeometric represettio + α, + α + β + P α,β x F α + i ; x,,,, Note tht this result lso holds for x. I view of the symmetry 3 we lso hve + β, + α + β + P α,β x F ; + x,,,,.... β + Note tht the hypergeometric represettio implies tht d + α + α + β + dx P α,β x α + + α + β + + α + β + ; x +, + α + β + F α + + α +, + α + β + F α + P α+,β+ x,,, 3, ; x

16 Aother cosequece of the hypergeometric represettio is tht the ledig coefficiet of the polyomil P α,β x equls + α + α + β + k α + + α + β +,,,,.... Now it c be show tht the Jcobi polyomils stisfy the orthogolity reltio x α + x β P α,β m xp α,β x dx α+β+ Γ + α + Γ + β + + α + β + Γ + α + β + δ m for α >, β > d m, {,,,...}. This c be show by usig the defiitio d itegrtio by prts. The vlue of the itegrl i the cse m c be computed by usig the ledig coefficiet d the writig the itegrl i terms of bet itegrl: { x α + x β P x} α,β dx + α + β + [ P α,β xd x +α + x +β] dx D P α,β x x +α + x +β dx Γ + α + β + Γ + α + β + d by usig the substitutio x t x +α + x +β dx x +α + x +β dx x +α + x +β dx,,,,... t +α t +β dt +α+β+ t +α t +β dt +α+β+ B + α +, + β + +α+β+ Γ + α + Γ + β + Γ + α + β + +α+β+ Γ + α + Γ + β +,,,, α + β + Γ + α + β + The Jcobi polyomils P α,β x stisfy the secod order lier differetil equtio x y x + [β α α + β + x] y x + + α + β + yx,,,,.... A geertig fuctio for the Jcobi polyomils is give by α+β R + R t α + R + t β P α,β xt, R xt + t. 6

17 Legedre The Legedre polyomils re orthogol o the itervl, with respect to the weight fuctio wx. They c be defied by mes of their Rodrigues formul: P x wx D [ wx x ] D [ x ],,,, This is the specil cse α β of the Jcobi polyomils:, + P x P, x F ; x,,,, Further we hve P x P x, P d P,,,,.... The ledig coefficiet of the polyomil P x equls k +!,,,,.... The orthogolity reltio is P m xp x dx + δ m, m, {,,,...}. 7 This c be show by usig the Rodrigues formul 5 d itegrtio by prts P m xp x dx P m xd [ x ] dx D P m x x dx, which vishes for m <. For m we hve D P x x dx k x dx! x dx. Filly we hve by usig the substitutio x t for,,,... x dx Hece we hve x + x dx t t dx + + Γ + Γ + B +, + + Γ + +!. {P x} dx! + +! This proves the orthogolity reltio 7.,,,,

18 I order to fid geertig fuctio for the Legedre polyomils we use the hypergeometric represettio 6 to fid P xt k + k x k t k k! k + k x k t k! k! k k k + k + k x k t +k k! k! k! x k t k k + t k! k! k! x k t k t k k! k! [ ] / k [x t] k t k t x t / k! t [ t x t ] / xt + t / This proves the geertig fuctio xt + t. P xt. 8 xt + t If we defie F x, t xt + t /, the we hve This implies tht t F x, t xt + t 3/ x + t x t xt + t 3/. xt + t F x, t x tf x, t. t Now we use 8 to obti xt + t P xt x t P xt. This c lso be writte s P xt x P xt + P xt + x P xt P xt + or equivletly + P + xt x + P xt + + P xt +. This leds to P x xp x d the three term recurrece reltio + P + x + xp x + P x,,, 3,

19 Chebyshev For x [, ] the Chebyshev polyomils T x of the first kid d the Chebyshev polyomils U x of the secod kid c be defied by T x cosθ d U x The orthogolity property is give by si + θ, x cos θ,,,, si θ d x / T m xt x dx x / U m xu x dx π π cosmθ cosθ dθ, m sim + θ si + θ dθ, m. Both fmilies of orthogol polyomils stisfy the three term recurrece reltio sice we hve d Note tht We lso hve sice P + x xp x P x,,, 3,..., T + x + T x cos + θ + cos θ cos θ cosθ xt x U + x + U x U x xu x si + θ + siθ si θ cos θ si + θ si θ T x U x, T x x d U x x. T x U x xu x,,, 3,..., si + θ cos θ siθ si θ si θ cosθ si θ xu x. cosθ T x. I order to fid geertig fuctio for the Chebyshev polyomils T x of the first kid, we multiply the recurrece reltio by t + d tke the sum to obti If we defie the we hve T + xt + x T xt + T xt +. F x, t T xt, t <, F x, t T xt T x xt [F x, t T x] t F x, t. 9

20 This implies tht xt + t F x, t T x + T xt xtt x + xt xt xt. Hece we hve the geertig fuctio T xt F x, t xt, t <. xt + t I the sme wy we hve for the Chebyshev polyomils U x of the secod kid: Gx, t U xt, t < where Hece we hve xt + t Gx, t U x + U xt xtu x + xt xt. U xt Gx, t, t <. xt + t This c be used, for istce, to prove tht T k xx k U x,,,,.... I fct, we hve for t < T k xx k t k T k xx k t T k xt k xt xt + t U xt. I similr wy we c prove tht P k xp k x U x,,,,..., T k xx t +k xt xt + t xt where P x deotes the Legedre polyomil. I fct, we hve for t < P k xp k x t P k xp k xt P k xp xt +k k P k xt k P xt xt + t U xt. xt + t xt + t