Complex Polynomials Orthogonal on the Semicircle

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1 Chapter Cmplex Plynmials Orthgnal n the Semicircle.. Intrductin GRADIMIR V. MILv ANOVIC, University fnis Suppse that X is a real linear space f functins, with an inner prduct if, g) :,X2 -+ R such that (a) if+ g, h) = if, h) + (g, h) (b) (a/, g) = a if, g) (c) if, g) = (g,j) (d) if,j) > 0, if,j) = 0 :> f= 0 where/, g, hex and a is a real scalar. (Linearity) (Hmgeneity) (Symmetry) (psitivity) If X is a cmplex linear space f functins, then the inner prduct if, g) maps,x2 int C and the requirement (c) is replaced by (c' if, g) = (g, f) (Hermitian Symmetry) where the bar designates the cmplex cnjugate. A system f plynmials {Pk}' where Pk(/) = Ik + tenns f lwer degree (k =0,,... ) and (Pk' p",) = 0 (k *m), (Pk' p",»o (k=m) is called a system f (mnic) rthgnal plynmials with respect t the inner prduct (.,. ). The mst cmmn type f rthgnality is with respect t the fllwing inner prduct (f, g)= J!(t)g(t)dJ.(t), R where da.{t) is a nnnegative measure n the real line R with cmpact r infinite supprt, 47

2 48 Cbapter fr which all mments I'k= J tkcu(t), k=o,,... R exist and are finite, and Jl > O. Then the (mnic) rthgnal plynmials {Pk} satisfy the fundamental three-term recurrence relatin () PHI (t) =(t - ak) Pk(t) - bkpk-l (t), P-l (t)=o, P (t)=, k= 0,, 2,..., where the cefficients ak and b k are given by ak= (tpk,pk) (k=o,,... ), (PkPk) bk = (Pk, Pk) (k =, 2,... ). (Pk-II Pk-t) We nte that b k > 0 fr k. The cefficient b in () is arbitrary, but the definitin is smetimes cnvenient. Typical examples f such plynmials are the classical rthgnal plynmials f Legendre, Ceby ev, Gegenbauer, Jacbi, Laguerre and Hermite. An ther type f rthgnality is the rthgnality n the unit circle with respect t the inner prduct 2n (/, g)= J /(ef")g(ef") dp.(o), These plynmials were intrduced and studied by SzegO []. Mnic rthgnal plynmials { <l>t} n the unit circle satisfy the recurrent relatin cp k + (z) = Z cpk (z) + cpk (0) Zk cpk (liz), k=o,,2,..., which is nt f the frm (). Fr mre details cnsult Nevai [2]. Similarly, we may cnsider rthgnal plynmials n a rectifiable curve r an arc lying in the cmplex plane (e.g. Gernimus [3], SzegO [4]). Cmplex rthgnal plynmials may als be cnstructed by means f duble integrals. Namely, intrducing the inner prduct by

3 Cmplex Plynmials Orthgnal n the Semicircle 49 (J, g)= J J J(z)g(z)w(z)dxdy. B fr a suitable psitive weight functin w where B is a bunded regin f the cmplex plane, a system f rthgnal plynmials can be generated (see Carleman [5] and Bchner [6])..2. Orthgnality n the Semicircle Recently Gautschi and [7] (see als [8]) intrduced a new type f rthgnality, the s-called rthgnality n the semicircle. The inner prduct is defined by () (J, g) = J J(z)g (Z)(iZ)- dz, r where r= {z eel z = 8, 0 S (}S tr}. Alternatively, () can be expressed in the frm (2) (J, g)= J f(e'')g (e'') do. r Ntice that this inner prduct des nt satisfy the cnditins (c' and (d). Namely, the secnd factr in (), i.e. (2), is nt cnjugated, s that this prduct des nt pssess Hermitian symmetiy; instead it has prperty (c). The crrespnding (mnic) rthgnal plynmials exist uniquely and satisfy a threeterm recurrence relatin f the frm () frm l., due t the prperty (zj, g) = if, zg). The general case f rthgnality with the cmplex weight functin w with respect t the inner prduct i.e. (J, g)= J!(z)g(z)w(Z)(iZ)-l dz, r r (3) (J, g)= J J(e'll)g(e'6) w(ei6) do, was cnsidered by Gautschi, Landau and [9]. Let w : (-, ) -. R+ be a weight functin, which can be extended t a functin z 4 w(z) which is regular in the half disc

4 ISO Chapter Tgether with (3) cnsider the inner prduct (4) [f. g) = J I(x)g (x)w(x)dx, - which is psitively definite and therefre generates a unique set f real (mnic) rthgnal plynmials {Pk}: [Pi' p",]=o fr k*,m and [Pi' p",]>o fr k=m (k, m=o,, 2,... ) On the ther hand, the inner prduct (3) is nt Hermitian; the secnd factr g is nt cnjugated and the integratin is nt with respect t the measure Iw(e j ')ld6. The existence f crrespnding rthgnal plynmials, therefre, is nt guaranteed. We call a system f cmplex plynmials {lrk} rthgnal n the semicircle if [lr i, '..] = 0 fr k *' m and ['." '..]*,0 fr k=m (k, m=o,, 2,...) where we assume that lrk is mnic f degree k. The existence f the rthgnal plynmials {lrk} can be established assuming nly that,. (5) Re(I,I)= Re lw(el9)cl6==o..3. Existence and Representatin f n;. Assume that the weight functin w is psitive n (-, ), regular in D+ and such that the integrals (3) and (4) frm.2 exist fr smth/and g (pssibly) as imprper integrals. We als assume that the cnditin (S) f.2 is satisfied. Let C 8' & > 0 dente the bundary f D + with small circular parts f radius & and centres at ±l spared ut and let P be the set f all algebraic plynmials. Then. by Cauchy's therem, fr any g e P we have () 0= J g(z)w(z)dz c.. J-. =( J + J + J )g(z)w(z)dz= J g (x) w (x) dx, r. C _l c

5 Cmplex Plynmials Orthgnal n the Semicircle 5 where rs and C.!;±l are the circular parts f Cs (with radii and & respectively). We assume that w is such that fr all g E P (2) lim J g(z)w(z)dz=o._0 C ±I (VgEP). Then, if 0 in (), we btain (3) 0= J g (z) w (z) dz = J g (z) w (z) dz + J g (x) w (x) dx, gep. C r -I I The (mnic, real) plynmials {Pk}' rthgnal with respect t the inner prduct (4) f.2, as well as the assciated plynmials f the secnd kind qk(z)= J Pk (Z)-Pk(X) w(x) dx z-x -I (k=o,, 2,... ), are knwn t satisfy a three-term recurrence relatin f the frm (4) (k=o,, 2,... ), where (5) Y_I = 0, Y = fr {Pk} and Y_I = -, Y = 0 fr {qk}. Dente by mk and Pk the mments assciated with the inner prducts (4) and (3) f.2, respectively where, in view f (5) (6) b = "'0. THEOREM. Let w be a weight functin. psitive n (-, ), regular in D+ = {z E Cllzl <, Imz> O} and such that (2) is satisfied and the integrals in (3) exist (pssibly) as imprper integrals. Assume in additin that If (7) Re(, ) = Re J w{e(9)do:t O. Then there exists a unique system f (mnic. cmplex) rthgnal plynmials {Irk} relative t the inner prduct (3) f.2. Denting by {Pk} the (mnic. rea/) rthgnal plynmials relative t the inner prduct (4) fl.2, we have

6 52 Chapter (8) where nn (z) =Pn (z) - i 0n-Pn-l (Z) (n=o,,2,... ), (9) = l'pn (0)+ iqn (0) n-l.. i l'pn-l (O)-qn-l (0) (n=o,,2,... ). Alternatively, (0) (n=o,, 2,... ); 0_ =-'0' where ak' bk are the recursin cefficients in (4) and P = (, ). In particular, all On are real (in fact, psitive) if an = 0 fr all n O. Finally, (n=,2,... ), Prf. Assume first that the rthgnal plynmials {Nk} exist. Putting in (3) we find g(z) = Nn(Z)Zk-l, I s: k < n 0= J nn (z) zk (iz)-l W (z) dz - i J nn (x),xk-l w (x) dx r-l ( :;;.k<n), and hence, upn expanding Nn in the plynmials {Pk}' (2) nn (z) = Pn (z) - i 0n_l Pn-l (z) (n=o,,2,... ). fr sme cnstants On-I. T determine these cnstants, put g (z)= [nn (z) - nn (0)] (iz)-l = {Pn(Z)-Pn (0) -i 0n-l Pn-l (Z)-Pn-l(O)},z z in (3) and use the first expressin fr g t evaluate the first integral, and the secnd t evaluate the secnd integral in (3). This gives ). Since (N n, ) = 0, (, ) = P, and using (2) with z = 0, we get (9) fr n. Nte that the denminatr in (9) (and the numeratr, fr that matter) des nt vanish, since Re P * 0

7 Cmplex Plynmials Orthgnal n the Semicircle 53 by (7) and Pk(O), qk(o) cannt vanish simultaneusly, {Pk} and {qk} being linearly independent slutins f (4). Fr n = 0, (9) yields, by virtue f (5), 0_ = JI. T shw the first relatin in (0), replace n by n + in (9) and use (4) fr z = 0, t btain _POPII+ (O)+iqll+ (0),- - ippii (O)-qll (0) P [-ailpii (O)-hIlPn_ (0)] + i [-all qll (O)-hll qll_ (0)] i POPII (O)-qll(O) iall [ippii(o)-qll (O)]-hll lpopii- (0) + iqll_ (0)] i P PII (O)-qll (0)' (- ). Using (9) with n =, (4) with k = 0 and (5), yields 8 P (-a) + ih m =Ia+-' 'P P since b = m (see (6». Therefre, (0) als hlds fr n = O. If all an = 0, then w is symmetric and we can prve that (see Therem in l.6) P = (,) = nw(o) > O. Hence, using (0) we cnclude that On is real. Cnversely, defining "n by (8) and (9), it fllws readily frm (3) that fr n 2, (nn, zk)= J r n,,(z)zk- w(z)dz=i J n,,(x)xk- w(x)dx=o _I and frm (3), (9) and (8) fr z = n that ("n' ) = 0 fr n l. Furthermre, (nil' nil) = J nn (z) zn w (Z)(iZ)- dz r J =[ nn(z)zn-w(z)dz=i f nil (x)xn-i w(x)dx r _I.. i J [PII(x)-iO Il _ PII_I(X)]xn-w(x)dx -I =0_ f - ' (x)w (x) dx,

8 IS4 Chapter prving (). We nte frm (8) that " (4) (p", )=io"_ (P"-h )=... = n (io._ )(I, ),.- and, similarly, which, applied repeatedly, gives (IS) Here, () has been used in the last step. Frm (4) and (IS) there fllws (6) the inversin f (8). When k = n, the empty prduct in (6) is t be interpreted as. EXAMPLE. w(z) = + z. Here, J.l = (I, ) = Ir + 2;, Re J.l *' 0, s that the rthgnal plynmials {Irk} exist. Fwhennre, b = m = 2, s that by (0) a = " (2n + X2n + 3) b = n(n+) " (2n+)2 n-4i 0 0 = 3 (2 - in), 3n+8i 0 =, S (4+ in) by (4) and (5), and by (8),

9 Cmplex Plynmials Orthgnal n the Semicircle 55 EXAMPLE 2. w(z) = r. Here " P- = J e2i8 do = 0 in 4 n ==Z2_.-- z-, 2 4+in 3 (4+ in) s that (7) is vilated and thus the plynmials {Irk} d nt exist, even thugh w(x) 0 n [-I, ] and the plynmials {Pk} d exist. It is easily seen that = 0 when k is even, s that 0n-I is zer fr n even, and undefmed fr n dd. Fr an explanatin f this example see Therem in Recurrence Relatin We assume that tr () Re(l, ) = Re J wk 8 )d8;t:. s that rthgnal plynmials {7rk} exist. Since (zf, g) = if, zg), it is knwn that they must satisfy a three-tenn recurrence relatin (2) 7'- (z) = 0, k =0,, 2,..., Using (8) f.3 and (2), fr k we get and substituting here fr zpk(z) and zpk_l (z) the expressins btained frm the basic recurrence relatin (4) f.3, yields [k + i «()k - ()k-l - a/] Pk (z) + [bk - {Jk - ()k-l (ak + iok-)]pk-l (z) + i [{Jk 0k-2 - bk- Ok-I] Pk-2 (z)= 0, k!;.. By the linear independence f the plynmials {Pk} we cnclude that ak +i(ok-ok_l- ak)=o, k!;.l, (3) k!;.i,

10 56 Chapter Frm the last equality in (3) and (0) f.3, we get (4) fr k > 2. The first equality in (3) gives (5) T verify that (4) als hlds fr k =, it suffices t apply the secnd relatin (3) fr k =, in cmbinatin with (0) f.3 and (5) fr k =. With a", Pk thus determined, the secnd equality in (3) is autmatically satisfied, as fllws easily frm (0) f.3. Finally, frm (z) = z- ja = PI (z) - jo =z - a - jo' we find Alternatively, by (0) f.3 we may write (5) and (6) as (7) We have therefre prved: THEOREM. Under the assumptin (). the (mnic. cmplex) plynmials {l'k} rthgnal with respect t the inner prduct (3) f.2 satisfy the recurrence relatin (2), where the cefficients ale> Pt are given by (5) (r (7» and (4), respectively, with the On defined in (9) (r (0» fll.3. By cmparing the cefficients f zk n the left and right f (2), we btain frm (5), (6) that (n )..5. Jacbi Weight We cnsider nw the case f the Jacbi weight functin

11 Cmplex Plynmials Orthgnal n the Semicircle 57 () +Z)II, a> -, P>I, where fractinal pwers are understd in terms f their principal branches. We first btain the existence f the crrespnding rthgnal plynmials {t4 a Jl)}. THEOREM. We have n (2) '0 =,00. {J) = J w<. II) (ei 8) do = 3: + i v. p. J W(O,: (x) dx, and hence Re P *" O. - Prf. Let C E, &> 0, be the cntur frmed by {JD+, with a semicircle f radius r abut the rigin spared ut. Then, by Cauchy's therem where rand ce are the circular parts f Ce (with radii and &, respectively). If (3) we get O=I'-i v. p. J W;X) dx-nw(o), - 0 in which prves (2). The fllwing therem is als prved in [9]: THEOREM2. We have 3:(fl.O)(z) ( ),,-(0. /I) ( ) " = - 3:". - Z, where frio dentes the plynmial frn with all cefficients cnjugated, i.e. n...(z) = frll(z)..6. Symmetric Weights and Gegenbauer Weights THEOREM. If the weight functin w. in additin t the assumptins stated in.3, satisfies () w( -z) =w(z) and w(o»o, then

12 58 (2),u = (, ) = nw (0), and the system %rthgnal plynmials {Kn} exists uniquely. Prf. Prceeding as in the prf f Therem frm.5, we find O=,u-l v.p.. J w(x) - -;-dx-nw(o), where the Cauchy principal value integral n the right vanishes because f symmetiy f w. This prves (2). Under the assumptin () we have ak = 0 fr all k 0 in (4) f.3. In this case the relatins (6), (5) and (4) f.4 reduce t. Furthermre, by (0) f.3 wherem =, 2,... In particular, fr the Gegenbauer weight (3) A> -/2, we have p" (z) = C" (z) - the mnic Gegenbauer plynmials - fr which as is well knwn, b =m = lrn r (i.+l/2) 0 r"' r(i.+l), b - k(k+2a-l) k-4(k+i.)(k+i.-l) ' - Therefre, by (0) f.3,

13 Cmplex Plynmials Orthgnal n the Semicircle 59 n(n+2a-l) ()n= -, n=l, 2, "" 4(n+A)(n+A-l) 8n - () _ r(a+l/2) 0- Vn r(a+l) i.e.. The crrespnding rthgnal plynmials can be represented in the frm :n;k(z)=c;(z)-i()k-l CLt(z) and their nrm is given by In particular, we have: The Legendre case A. = 2: () = ( r «k + 2)/2) )2 k 0; k r«k+l)/2), 2 The teby ev case A. = 0: 0 0 =, Ok = 2, k ; 3 The teby ev case f the secnd kind A. = : Ok = 2, k O. The Legendre case is cnsidered in detail in [7], and Gegenbauer case, including varius applicatins t numerical analysis, in [0]..7. The zers f n;,(z) It fllws frm (2) f.4 that the zers f 7rn(z) are the eigenvalues f the (cmplex, tridiagnal) matrix J= n iag 0 PI ia, p, iaz 0 Pn-I ian_,

14 60 chapter The elements f I n are easily cmputed using (6), (5), (4) f.4 and (0) f.3, since the recursin cefficients ale> bk fr the rthgnal plynmials {Pk} are knwn. If the weight w is symmetric, then Pk = LI is psitive, and I n can be transfrmed int a real matrix. Indeed, a similarity transfrmatin with the diagnal matrix transfrms the cmplex matrix I n int the real nnsymmetric tridiagnal matrix with eigenvalues Jv = -isv. Using the EISPACK subrutine HQR (see [» we can evaluate all the eigenvalues Jv (v=,...,n) f An' and then all the zers sv= ijv (v=,...,n) f 7rn(z). A therem n the distributin f zers f 7rn(z) in the case when the weight functin w is symmetric, i.e. w(-z) = w(z) and w(o) > 0 is prved in [9]. We shall qute here a particular, but imprtant, result regarding the distributin f zers f the plynmials n;!(z), rthgnal with respect t the cmplex Gegenbauer weight w(z) = (l-z 2 r" 2, A. > -2. THEOREM.f A. > -2, all the zers f are simple and fr n 2 they belng t the upper unit half disc D+ = {z E Cllzl <, Imz> O}. REMARK. The prblem f distributin f zers fr general weight functins is nt slved. The case f Jacbi weights (see () f.5) is particularly interesting. Numerical experiments indicate that all the zers belng t the half disc D +. REMARK 2. Fr the Gegenbauer case a secnd rder linear differential equatin is fund in [9] whse particular slutin is the plynmial 7r:(z). Applicatins f thse plynmials t numerical integratin and numerical differentiatin f analytic functins are given in [0). REFERENCES. G. SzegO: Ober Trignmetrische und Harmnische Plynme. Math. Ann. 79(98),

15 Cmplex Plynmials Orthgnal n the Semicircle 6 2. P. Nevai: Geza Freud, Orthgnal Plynmials and Christffel Functins. A Case Study. J. Apprx. Thery 48(986), Va. L. Gernimus: Plynmials Orthgnal n a Circle and Interval. Pergamn, Oxfrd G. SzegO: Orthgnal Plynmials. 4th ed. Amer. Math. Sc. Cllq. Publ. Vl. 23, Amer. Math. Sc., Prvidence, R T. Carlemann: Ober die Apprximatin Analytischer Funktinen durch lineare Aggregate vn vrgegebenen Ptenzen. Ark. Mat. Astrnm. Fys. 7( 92223), S. Bchner: Ober rthgnale Systeme Analytischer Funktinen. Math. Z. 4(922), W. Gautschi, G. V. Milvanvic: Plynmials Orthgnal n the Semicircle. J. Apprx. Thery 46(986), W. Gautschi, G. V. Milvanvic: Plynmials Orthgnal n the Semicircle. Rend. Sem. Mat. Univ. Plitec. Trin (Special Functins: Thery and Cmputatin), 985, W. Gautschi, H. J. Landau, G. V. Milvanvic: Plynmials Orthgnal n the Semicircle. Cnstr. Apprx. 3(987), G. V. Milvanvic: Cmplex Orthgnality n the Semicircle with respect t Gegenbauer Weight: Thery and Applicatins. In: Tpics in Mathematical Analysis (Th. M. Rassias, ed.), Wrld Scientific Singapre 989, pp B. T. Smith et al.: Matrix Eigensystem Rutines - EISPACK Guide. Lect. Ntes Cmp Sci. Vl. 6, Springer-Verlag, New Yrk 974.