Connection of Semi-integer Trigonometric Orthogonal Polynomials with Szegő Polynomials

Transcription

1 oectio of Semi-iteger Trigoometric Orthogoal Polyomials with Szegő Polyomials Gradimir V. Milovaović, leksadar S. vetković, ad Zvezda M. Marjaović Departmet of Mathematics, Faculty of Electroic Egieerig, Uiversity of Niš, P.O. Box 73, 8000 Niš, Serbia bstract. I this paper we ivestigate coectio betwee semi-iteger orthogoal polyomials ad Szegő s class of polyomials, orthogoal o the uit circle. We fid a represetatio of the semi-iteger orthogoal polyomials i terms of Szegő s polyomials orthogoal o the uit circle for certai class of weight fuctios. Itroductio Let us deote by T +/ the liear spa of trigoometric fuctios cos x/, si x/, cos( + /)x, si( + /)x,..., cos( +/)x, si( +/)x. The elemets of T +/ are called trigoometric polyomials of semi-iteger degree. For the coveiece sake we defie T / = {0}. learly,t +/ is a liear space of dimesio +.Weitroduceierproductby (f,g) = f(x)g(x)w(x)dx, f, g T +/, () where w is a o-egative weight fuctio o (, π], which equals zero oly o asetoflebesguemeasurezero. Next, we defie the followig set T a,b +/ = a cos( +/)x + b si( +/)x + T /, where a, b R are fixed with the property a + b > 0. The case a = b =0is ot iterestig sice i that case T 0,0 +/ = T /. Givetheierproduct(), we ca pose a questio of fidig T a,b +/,suchthat (x)t(x)w(x)dt =0, t T /. It turs out that this problem has a uique solutio. The authors were supported i part by the Serbia Miistry of Sciece ad Evirometal Protectio (Project: Orthogoal Systems ad pplicatios, grat umber #44004) ad the Swiss Natioal Sciece Foudatio (SOPES Joit Research Project No. IB New Methods for Quadrature ). T. Boyaov et al. (Eds.): NM 006, LNS 430, pp , 007. c Spriger-Verlag Berli Heidelberg 007

2 oectio of Semi-iteger Trigoometric Orthogoal Polyomials 395 Lemma. There exist a uique T a,b +/,suchthat Proof. y polyomial t T a,b +/ (x)t(x)w(x)dx =0, t T /. ca be represeted as t(x) =a cos(+/)x+b si(+/)x+ [a k cos(k +/)x+b k si(k +/)x]. I order to be orthogoal to T /,itscoefficietshavetosatisfythefollowig system of liear equatios [ ak (cos(k +/)x, cos(l +/)x)+b k (si(k +/)x, cos(l +/)x) ] = (a cos( +/)x + b si( +/)x, cos(l +/)x), [ ak (cos(k +/)x, si(l +/)x)+b k (si(k +/)x, si(l +/)x) ] = (a cos( +/)x + b si( +/)x, si(l +/)x), for l =0,...,. The solutio of this system is uique sice its matrix is a Gram matrix formed by liearly idepedet vectors (see [, p. 4]). From ow o, we deote by T a,b +/ the trigoometric polyomial orthogoal to T /.Whewewattoemphasizethedepedeceoa ad b, wewrite for T a,b +/.FromtheproofofLemmaitisclearthatwehave (x) =a,0 + b0,. () I [], we used otatio =,0 ad S = 0, for obvious reasos. osider the followig quadrature rule w(x)t(x)dx = w k p(x k ), t T, where T deotes liear spa of trigoometric fuctios, cos x, six,...,cos x, si x. I[4]ad[]itisprovedthatsuchquadratureruleexists,ithaspositive weights w k, k =0,...,, aditsodesx k, k =0,...,, arezerosofthe polyomial T a,b +/,orthogoaltot /, wherea, b R, a + b > 0, are arbitrary. We ote that a basis of T a,b +/ { k, b,a k is the set k =0,...,, a + b > 0, a,b R} { }.

3 396 G.V. Milovaović,.S. vetković, ad Z.M. Marjaović This is obvious sice cos(k +/)x + t(x) = si(k + /)x + t(x) = a b a + b a,b k a + b b,a k, b a a + b a,b k + a + b b,a k, where t T k /, k =0,...,. I [], it is proved that the sequeces of polyomials followig five-term recurrece relatios ad S satisfy the =(cosx α ) α β S β S, (3) S =(cosx δ ) S δ S γ γ. Usig relatios (), we ca prove easily the followig Lemma. satisfy the followig five term recur- Lemma. Polyomials rece relatios ad b,a b,a =(cosx α ) α β b,a β b,a, =(cosx δ ) b,a δ b,a γ a,b γ a,b, where α = a α + ab(β + γ )+b δ a + b, α = a α + ab(β + γ )+b δ a + b β = a β ab(α δ ) b γ a + b, β = a β ab(α δ ) b γ a + b δ = b α ab(β + γ )+a δ a + b, δ = b α ab(β + γ )+a δ a + b, γ = a γ ab(α δ) b β a + b, γ = a γ ab(α δ) b β a + b Proof. Just use coectio = aa,b b b,a i (3), ad solve liear system for + a b,a a + b, S = ba,b a + b, oectio to the SzegőPolyomials ad b,a. I the rest of this paper we shall eed the followig lemma which gives a factorizatio of the positive trigoometric polyomials. Recall that T is liear space spaed by the set of trigoometric fuctios cos kx, si kx, k =0,...,.By P, N 0,wedeotethesetofallalgebraicpolyomialsofdegreeatmost. The ext lemma (i a slightly differet formulatio) ca be foud i [3, p. 4].

4 oectio of Semi-iteger Trigoometric Orthogoal Polyomials 397 Lemma 3. Let t T, N, beatrigoometricpolyomialofexactdegree, whichisstrictlypositiveotheiterval(, π]. Thethereexistauique (up to a multiplicative costat of modulus oe) algebraic polyomial H P of exact degree, suchthat t (x) =e ix H (e ix )H (e ix ), where H (z) =z H (/z), (4) ad all the zeros of H are of modulus smaller the oe. Proof. Let t (x) = the we ca expad ( ak cos kx + b k si kx ), a + b > 0, t (x) =e ix T (e ix )=e ix ( ak ib k where T is a algebraic polyomial of exact degree. For the polyomial T we have T (z) =z T (/z) =z ( ak +ib k e i(+k)x + a k +ib k e i( k)x), z +k + a k ib k ) z k = T (z). s a cosequece we have that if T (z) =0,thealsoT (/z) =0.Notethat T (0) = (a +ib )/ 0.Hece,T has zeros of modulus smaller the oe ad zeros of modulus bigger the oe. Notice that T ca ot have a zero e ix, x R, siceithatcaset (x) =e ix T (e ix )=0,whichisimpossibleaccordig to the assumptios of the theorem. Deote by z k, k =,...,,thezerosoft of modulus smaller the oe, the we have T (z) = a ib (z z k )(z /z k ). To esure that H is of exact degree with all its zeros lyig iside the uit circle, we set H (z) = (z z k ), with some. The ad obviously H (z) =z k= (/z z k )= k= H (z)h (z) = k= ( z k ) ( z k ) k= (z /z k ), k= (z z k )(z /z k ). k=

5 398 G.V. Milovaović,.S. vetković, ad Z.M. Marjaović I order to get the desired represetatio, there must hold = a ib k= ( z k). (5) Hece, it remais to prove that the quatity o the right is positive. We have t (x) =e ix T (e ix )= a ib (e ix z k )( e ix /z k ) k= = a ib k= ( z k) e ix z k, hece it is positive. Sice we imposed oly coditio o the modulus of, we are free to choose its argumet. Now, we are ready to prove the followig theorem. Theorem. Let t l T l, l N 0,beatrigoometricpolyomialofexactdegree l, strictlypositiveo(, π]. The,for l, thepolyomial T a,b +/, orthogoal with respect to the weight fuctio w(x) =/t l (x), isgiveby (x) = a ib k= e i( l+/)x h l (e ix )+ a +ib e i( l+/)x h l (e ix ), where h l is the moic versio of the polyomial H l from Lemma 3. The coefficiets i five term recurrece terms are give by α = β = γ = δ =0, l +, β = γ =0, α = δ =, l +. Proof. To prove orthogoality of to T /,itsufficestoestablishitsorthogoality to the set e i(k+/)x, k =,...,0,...,. Withc =(a ib)/, we have e i(k+/)x t l (x) = ic z k (cz l+ h l (z)+cz +l h l (z )) dx = i H l (z)h l (z ) z +k Hl (z)dz ic z +k+l dz =0, z = e ix, H l (z) (x) where deotes the uit circle, ad is the leadig coefficiet i H l. The first itegral equals zero, due to the fact that, for k =,...,0,..., the itegrad does ot have sigularities iside the uit circle.the secod itegral equals zero sice for k =,...,0,..., the itegrad does ot have sigularities outside the uit circle ad it is of order at least z as z teds to ifiity. Usig similar methods as i [], it ca be proved that the coefficiets i the five term recurrece relatios are give uiquely as the solutios of the followig systems of liear equatios dz

6 oectio of Semi-iteger Trigoometric Orthogoal Polyomials 399 J a,b, j = αj Ia,b j + β j I j, J j, = γ j I a,b j + δ j I j, J, j = α j I j + β j I b,a j, J b,a, j = γj I j + δ j I b,a j, for j =,, ad >, where J a,b, =(cosx, ), J a,b, = Ia,b, I a,b j =(a,b j,a,b j )= a,b j, I j =( j, b,a j ), J, =(cosx, b,a ), J, = J, = I. Next we calculate the orm of the polyomial k.wehave = I a,b = ( (x)) t (x) ic dx = ic z l h l (z) Hl (z) dz i c dz z z ( l) h l (z ) dz = 4π c H l (z) = π a + b, where the first ad the third itegrals are equal to zero sice their itegrads are aalytic iside ad outside the uit circle, respectively, with itegrad i the third itegral beig of order at least z at ifiity. For the itegral J,, a,b wehave J,= a,b cosx( t l (x) (x)) dx = ic ic (z + z )z l h l (z) Hl (z) i c z + z dz (z + z )z ( l) h l (z ) dz =0, H l (z) usig the same argumetatio as i calculatig the previous itegral. Next, deotig d = (b +ia)/, we have I = (x) b,a t l (x) (x) dx = icd z l h l (z) Hl (z) dz icd i(cd + cd) dz z z ( l) h l (z ) dz =0, H l (z) where the first ad the third itegrals are zero by the same argumets as above, ad for the secod oe we have Fially, we have J, = cd + cd == ab ab i(a + b )+ab ab +i(a + b ) 4 cosx i(cd + cd) (x) b,a (x) dx = icd t l (x) z + z dz icd =0. (z + z )z l h l (z) Hl (z) dz z ( l) h l (z ) dz =0. H l (z)

7 400 G.V. Milovaović,.S. vetković, ad Z.M. Marjaović Usig the systems of liear equatios for the five term recurrece coefficiets, we get J a,b, =0= α I a,b + β I = α, J, =0= α I + β I b,a, = β b,a, J a,b, = Ia,b = a,b = α I a,b + β I = α, J, = I =0= α I + β I b,a J, =0= γ I a,b + δ I = γ, J b,a, =0= γ I + δ I b,a = δ b,a, = β b,a, J, = I =0= γ Ia,b + δ I = γ a,b, J b,a, = I b,a = b,a = γ I + δ I b,a = δ b,a. Sice the orms of the polyomials from zero, we get what is stated. ad b,a are the same, ad differet s we ca see, we established acoectiowithszegő s polyomials. Recall that Szegő s polyomials (see [3, p. 87]) are defied to be orthogoal o the uit circle with respect to the ier product (p, q) = π p(e ix )q(e ix )w(x)dx, p, q P. I [3, p. 89], it ca be foud that for the special type of weights w(x) =/t l (x), t l T l,wheret l is strictly positive o (, π], Szegő moicpolyomialscabe expressed as φ (e ix )=e i( l)x h l (e ix ), l. Hece, we have established the followig Theorem. Theorem. The trigoometric polyomial T a,b +/, l, orthogoal with respect to the strictly positive weight fuctio w(x) =/t l (x), t l T,ca be represeted as ib (x) =a e ix/ φ (e ix )+ a +ib e ix/ φ (e ix ), l, where φ is the respective Szegő polyomialorthogoalotheuitcircle. Moreover, the orm of the polyomial, l, isgiveby ( = π(a + b )exp ) log t l (x)dx. π Proof. We eed to prove oly the statemet about the orm. I [3, p ], it is prove that the orm of the moic Szegő polyomialisgiveby ( φ =exp ) log t l (x)dx. π ccordig to the proof of Theorem, we have = π(a +b ) φ.

8 Refereces oectio of Semi-iteger Trigoometric Orthogoal Polyomials 40. Milovaović, G. V., vetković,. S., Staić, M. P.: Trigoometric quadrature formulae ad orthogoal systems (submitted). Milovaović, G. V., Djordjević, R. Ž.: Liear lgebra. Faculty of Electroical Egieerig, Niš (004)[ISerbia] 3. Szegő, G.: Orthogoal Polyomials. mer. Math. Soc. olloq. Publ. 3, 4thed.. mer. Math. Soc., Providece R. I. (975) 4. Turetzkii,. H.: O quadrature formulae that are exact for trigoometric polyomials. East J. pprox. (005) (Eglish traslatio from Ucheye Zapiski, Vypusk (49), Seria math. Theory of Fuctios, ollectio of papers, Izdatel stvo Belgosuiversiteta imei V.I. Leia, Misk (959) 3 54)