Orthogonal Rational Functions, Associated Rational Functions And Functions Of The Second Kind

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1 Proceedings of he World Congress on Engineering 2008 Vol II Orhogonal Raional Funcions, Associaed Raional Funcions And Funcions Of The Second Kind Karl Deckers and Adhemar Bulheel Absrac Consider he sequence of poles A α 1, α 2,..., and suppose he raional funcions ϕ j wih poles in A form an orhonormal sysem wih respec o a Hermiian posiive-definie inner produc. Furher, assume he ϕ j saisfy a hree-erm recurrence relaion. Le he raional funcion ϕ (1) wih poles in α 2, α 3,... represen he associaed raional funcion of ϕ j of order 1; i.e. he ϕ (1) do saisfy he same hree-erm recurrence relaion as he ϕ j. In his paper we hen give a relaion beween ϕ j and ϕ (1) in erms of he so-called raional funcions of he second kind. Nex, under cerain condiions on he poles in A, we prove ha he ϕ (1) form an orhonormal sysem of raional funcions wih respec o a Hermiian posiive-definie inner produc. Finally, we give a relaion beween associaed raional funcions of differen order, independen of wheher hey form an orhonormal sysem. Keywords: Orhogonal raional funcions, associaed raional funcions, raional funcions of he second kind, hree-erm recurrence relaion, Favard heorem. 1 Inroducion Le φ j denoe he polynomial of degree j ha is orhogonal wih respec o a posiive measure µ on a subse S of he real line. Furher, suppose he orhogonal polynomials (OPs) φ j are monic (i.e. hey are of he form φ j (x) x j +...) and saisfy a hree-erm recurrence relaion given by φ 1 (x) 0, φ 0 (x) 1, φ j (x) (x α j )φ j 1 (x) β j φ j 2 (x), j 1. Le he monic polynomial of degree j k denoe he associaed polynomial (AP) of order k 0, wih j k. By definiion, hese APs are he polynomials generaed by he hree-erm recurrence relaion given by 1 (x) 0, φ(k) 0 (x) 1, (x) (x α j) (j 1) k (x) β j (j 2) k (x), j k + 1. The work is parially suppored by he Fund for Scienific Research (FWO), projec RAM: Raional modeling: opimal condiioning and sable algorihms, gran #G , and by he Belgian Nework DYSCO (Dynamical Sysems, Conrol, and Opimizaion), funded by he Ineruniversiy Aracion Poles Programme, iniiaed by he Belgian Sae, Science Policy Office. The scienific responsibiliy ress wih he auhors. Deparmen of Compuer Science, Kaholieke Universiei Leuven, Heverlee, Belgium. Karl.Deckers@cs.kuleuven.be Deparmen of Compuer Science, Kaholieke Universiei Leuven, Heverlee, Belgium. Adhemar.Bulheel@cs.kuleuven.be Noe ha his way he monic APs of order 0 and he monic OPs are in fac he same. The following relaions exis beween monic APs of differen order (see e.g. 7) and (x) (x α k+1)φ (k+1) j (k+1) (x) β k+2φ (k+2) j (k+2) (x), j k + 1 (1) (x) φ(l) j l (x)φ(k) l k (x) β l+1φ (l+1) j (l+1) (x)φ(k) (l 1) k (x), k + 1 l j 1. (2) From he Favard heorem i follows ha he APs of order k form an orhogonal sysem wih respec o a posiive measure µ (k) on S. Therefore, anoher relaion exiss beween he APs of order k 1 and k in erms of polynomials of he second kind: (x) S φ (k 1) j (k 1) () φ(k 1) j (k 1) (x) dµ (k 1) (). (3) x Orhogonal raional funcions (ORFs) on a subse S of he real line (see e.g. 2, 5, 6 and 1, Chap. 11) are a generalisaion of OPs on S in such a way ha hey are of increasing degree wih a given sequence of complex poles, and he OPs resul if all he poles are a infiniy. Le ϕ j denoe he raional funcion wih j poles ouside S ha is orhogonal wih respec o a posiive measure µ on S. Under cerain condiions on he poles, hese ORFs do saisfy a hree-erm recurrence relaion as well. Consequenly, associaed raional funcions (ARFs) can be defined based on his hree-erm recurrence relaion. Furhermore, in 1, Chap. 11.2, he raional funcion of he second kind ϕ 1 j of ϕ j is defined similarly as in (3); i.e. ϕ 1 j (x) S ϕ j () ϕ j (x) dµ(). (4) x The aim of his paper is o generalise he relaions for APs, given by (1) (3), o he case of ARFs. Bu firs, we sar wih he necessary heoreical background in he nex secion. 2 Preliminaries The field of complex numbers will be denoed by C and he Riemann sphere by C C. For he real line we use

2 Proceedings of he World Congress on Engineering 2008 Vol II he symbol R, while he exended real line will be denoed by R R. Furher, we represen he posiive real line by R + x R : x 0. If he value a X is omied in he se X, his will be represened by X a ; e.g. C 0 C \ 0. Le c a + ib, where a, b R, hen we represen he real par of c C by Rc a and he imaginary par by Ic b. Given a sequence A j α 1, α 2,..., α j C 0, we define he facors x Z l (x), l 1, 2,..., j, 1 x/α l and producs b 0 (x) 1, b l (x) Z l (x)b l 1 (x), l 1, 2,..., j, or equivalenly, b l (x) xl π l (x), π l(x) l (1 x/α i ), π 0 (x) 1. i1 The space of raional funcions wih poles in A j is hen given by L j spanb 0 (x), b 1 (x),..., b j (x). We will also need he reduced sequence of poles A α k+1, α k+2,..., α j, where 0 k j, and he reduced space of raional funcions wih poles in A given by where L spanb k\k (x), b (k+1)\k (x),..., b (x), for l k and π l\k (x) b l\k (x) b l(x) b k (x) xl k π l\k (x), l ik+1 (1 x/α i ), π l\l (x) 1. In he special case in which k 0 or k j, we have ha A j\0 A j and L j\0 L j, respecively A j\j and L j\j L 0 C. We will assume ha he poles in A j are arbirary complex or infinie; hence, hey do no have o appear in pairs of complex conjugaes. We define he subsar conjugae of a funcion f(x) L by f (x) f(x). Consider an inner produc ha is defined by he linear funcional M: f, g Mfg, f, g L. We say ha M is a Hermiian posiive-definie linear funcional (HPDLF) if for every f, g L i holds ha f 0 Mff > 0 and Mfg Mf g. Furher, le µ 0 be defined as µ 0 M1 R + 0, and suppose here exiss a sequence of raional funcions ϕ j, wih ϕ j L j \ L j 1, so ha he ϕ j form an orhonormal sysem wih respec o M. Le α 0 C 0 be arbirary bu fixed in advance. Then he orhonormal raional funcions (ORFs) ϕ j p j π j are said o be regular for j 1 if p j (α j 1 ) 0 and p j (α j 1 ) 0. A zero of p j a means ha he degree of p j is less han j. We now have he following recurrence relaion for ORFs. For he proof, we refer o 5, Sec. 2 and 3, Sec. 3. Theorem 2.1. Le E 0 C 0, α 1 R 0 and α 0 C 0 be arbirary bu fixed in advance. Then he ORFs ϕ j, j n 2, n 1, n, wih n 1, are regular iff here exiss a hree-erm recurrence relaion of he form ϕ n (x) Z n (x) E n + F n ϕ n 1 (x) wih E n 0 and Z n 2 (x) ϕ n 2(x) C n E n + F n /Z n 1 (α n 1 ) E n 1 0. The iniial condiions are ϕ 1 (x) 0 and ϕ 0 (x) where η is a unimodular consan ( η 1)., (5) η µ0, Le L denoe he associaed raional funcion (ARF) of ϕ j of order k; i.e., j k + 1, k + 2,..., is generaed by he same recurrence relaion as ϕ j wih iniial condiions (k 1)\k(x) 0 and ϕ(k) k\k (x) κ(k) 0. Noe ha in he special case in which k 0, we have ha ϕ (0) j\0 ϕ j. In he remainder of his paper we will assume ha is of he form wih (x) p(k) (x) π (x), p (k) (x) κ(k) x +..., κ (k) C and κ (k) 0 0. We now have he following Favard heorem. For he proof, we refer o 4. Theorem 2.2 (Favard). Le be a sequence of raional funcions, and assume ha (A1) α k 1 R 0 and α j C 0, j k, k + 1,..., (A2), j k + 1, k + 2,..., is generaed by a hree-erm recurrence relaion of he form given by Equaion (5), (A3) L \ L (j 1)\k, j k + 1, k + 2,..., and k\k C 0,

3 Proceedings of he World Congress on Engineering 2008 Vol II (A4) Le ˆF j F j /E j, wih E j E (k) and F j F (k). Then ˆF j < and Iα j 2 α j 2 2 Iα j α j 2 E j 1 2 E j 2 Iαj 1 α j 1 2 ˆF j 2 I ˆF j E j Iα j 1 α j 1 2 Iα j 2 α j 2 2, j k + 1, k + 2,..., (A5) max 0, 4 Iα j α j Iα j 1 2 α j 1 2 < E j 2 <, j k, k + 1,..., (A6) C j E j 1 E j + F j /Z j 1 (α j 1 ) 0, wih C j C (k), j k + 1, k + 2,.... Then here exiss a HPDLF M (k) so ha f, g M (k) fg fg dµ (k) defines a Hermiian posiive-definie inner produc for which he raional funcions form an orhonormal sysem. In he remainder we will assume ha he sysem of ORFs ϕ j saisfies every assumpion in Theorem 2.2. This way, if α k 1 R 0, i is sufficien o prove ha / L (j 1)\k for j k + 1, k + 2,..., so ha he ARFs form an orhonormal sysem wih respec o a HPDLF M (k). If α k 1 R 0 and condiion (A3) is saisfied as well, we le M (k) 1 µ (k) 0 κ (k) Associaed raional funcions Suppose he ARFs of order k 1 0 form an orhonormal sysem wih respec o a HPDLF, and le Φ be given by Φ (x, ) (1 /α k 1 ) (x). (6) Then we define he raional funcions of he second kind ψ by ψ (x) (1 x/α k ) Φ (, x) Φ (x, ) x δ j,k 1 R k 1, j k 1, (7) where δ j,k 1 is he Kronecker Dela and R k 1 S κ (k 1) 0 α k 1 1. (8) Noe ha his definiion is very similar o, bu no exacly he same as he one given before in (4). We will hen prove ha he ψ saisfy he same hree-erm recurrence relaion as wih iniial condiions ψ (k 1)\k(x) 0 and ψ k\k (x) E k 1 C k /κ (k 1) 0 0. Firs, we need he following lemma. Lemma 3.1. Le ψ, wih j k 1 0, be defined as before in (7). Then i holds ha 0, j k 1 ψ (x) E k 1 C k /κ (k 1) 0 0, j k, while ψ L for j > k. Proof. Define q j (k 2) by q j (k 2) (x) (1 x/α k 1 )π (x) For j k i hen follows from (6) and (7) ha 1 ψ (x) π (x) 1 x ()q j (k 2)(x) (1 /α k 1 )p (k 1) j (k 1) (x) Furher, wih we have ha so ha a (k) j (k 1) () For j k we find ha κ (k) 0 Noe ha j (k 1) i0 π (x) π (x) c j,k lim x x, a (k) i c j,k 1 α k 1 0, ψ (x) p(k) (x) π (x) L. () k\(k 1) ()q 2(x) (1 /α k 1 )p (k 1) lim q 2(x) x α k 1 x x(1 /x) k\(k 1) () 1 /x x i. (9) () 1 (x). 0,

4 Proceedings of he World Congress on Engineering 2008 Vol II so ha κ (k) 0 lim x α k 1 1 /αk 1 1 /x κ (k 1) 0 2 (k 1) ϕk\(k 1) lim (x) x α k 1 1 lim x α k 1 κ (k 1) 0 E k + F k Z k 1 (x) E k + F k /Z k 1 (α k 1 ) /κ (k 1) 0. p (k 1) 1 (x) x Finally, in he special case in which j k 1, we have ha κ (k 1) 0 Φ(k 1)\(k 1) (, x) Φ (k 1)\(k 1) (x, ) x (1 x/αk 1 ) (1 /α k 1 ) R k 1, x where R k 1 is given by (8). The following heorem now shows ha hese ψ do saisfy he same hree-erm recurrence relaion as he ϕ j. Theorem 3.2. Le ψ be defined as before in (7). The raional funcions ψ, wih j n 2, n 1, n and n k + 1, hen saisfy he hree-erm recurrence relaion given by ψ n\k (x) Z n (x) E n + F n ψ (n 1)\k (x) Z n 2 (x) ψ (n 2)\k(x) The iniial condiions are ψ (k 1)\k (x) 0 and ψ k\k (x) E k 1 C k /κ (k 1) (10) Proof. Firs noe ha he ARFs, wih j n 2, n 1, n, do saisfy he hree-erm recurrence relaion given by (10), and hence, so do he Φ n\(k 1). We now have ha ψ n\k (x) 1 x/α k E n 1 x Z n ()Φ (n 1)\(k 1) (, x) Z n (x)φ (n 1)\(k 1) (x, ) + F n 1 x Zn () Z n 1 () Φ (n 1)\(k 1)(, x) Z n(x) Φ (n 1)\(k 1)(x, ) 1 x Zn () Z n 2 () Φ (n 2)\(k 1)(, x) Z n(x) Z n 2 (x) Φ (n 2)\(k 1)(x, ), ψ n\k (x) Z n (x) E n + Z n 2 (x) ψ (n 2)\k(x) F n ψ (n 1)\k (x) + fn (x, ) x + δ n,k+1 R k 1 C k+1 (1 x/α k ) Z k+1(x) Z k 1 (x) where f n (x, ) (1 x/α k )g n (x, ) and g n (x, ) is given by g n (x, ) E n Z n () Z n (x)φ (n 1)\(k 1) (, x) Zn () + F n Z n 1 () Z n(x) Φ (n 1)\(k 1) (, x) Noe ha Zn () Z n 2 () Z n () Z n (x) Z n () Z n 1 () Z n(x) Z n () Z n 2 () Z n(x) Z n 2 (x) so ha f n (x, ) x where h n () Z n(x) Z n 2 (x), Φ (n 2)\(k 1) (, x). ( x) (1 /α n )(1 x/α n ) ( x)/z n 1 (α n ) (1 /α n )(1 x/α n ) ( x)/z n 2 (α n ) (1 /α n )(1 x/α n ), Z n (x) (1 x/α k ) (1 /α n ) 1 h n () Z k 1 (x) ( Z n (x) (1 x/α k ) 1 + Z ) n() h n (), Z k 1 (x) α n E n (n 1)\(k 1) () + I clearly holds ha Furher, noe ha and Hence, F n Z n 1 (α n ) ϕ(k 1) (n 1)\(k 1) () + C n Z n 2 (α n ) ϕ(k 1) (n 2)\(k 1) (). δ n,k+1 C k+1 h n () κ (k 1) 0 Z k 1 (α k+1 ). Z n () Z n 2 (α n ) Z n() Z n 2 () 1 Z n () Z n 1 (α n ) Z n() Z n 1 () 1. Z n ()h n () n\(k 1) () F n (n 1)\(k 1) () C n (n 2)\(k 1) (),

5 Proceedings of he World Congress on Engineering 2008 Vol II so ha Z n ()h n () δ n,k+1c k+1. α n κ (k 1) 0 α k+1 Consequenly, we have ha fn (x, ) x which ends he proof. δ n,k+1 R k 1 C k+1 (1 x/α k ) Z k+1(x) Z k 1 (x) The nex heorem direcly follows from Lemma 3.1 and Theorem 3.2. Theorem 3.3. Le ψ be defined as before in (7). These ψ are he ARFs of order k wih iniial condiions (k 1)\k(x) 0 and k\k (x) E k 1C k /κ (k 1) 0 0. In he above lemma and heorems we have assumed ha he ARFs form an orhonormal sysem wih respec o a HPDLF. The assumpion cerainly holds for k 1, and hence, he ARFs ϕ (1) are he raional funcions of he second kind of he ORFs ϕ j. The nex quesion is hen wheher he ARFs ϕ (1) form an orhonormal sysem wih respec o a HPDLF M (1). Therefore, we need he following lemma. Lemma 3.4. Le he ARFs (7). Then he leading coefficien K (k) of b in he expansion of b k\k,..., b, is given by K (k) K(k 1) j (k 1) Proof. Noe ha he leading K (k) Thm. 3.1) K (k) lim (k) ϕ x α j (x), of order k be defined by, i.e. he coefficien wih respec o he basis 1 /αk 1 1 /α j b (x) lim, j k. is given by (see also 3, p (k) (x) x α j x. Furher, le q j (k 2) be defined as before in Lemma 3.1. Clearly, for j k i hen holds ha lim q j (k 2)(x) k\(k 1) () x α j 1 /x 0. x j (k 1) So, from (9) we deduce ha K (k) lim (k 1) pj (k 1) (x) x α j x j (k 1) K (k 1) j (k 1) 1 /αk 1 1 /x 1 /αk 1. 1 /α j This proves he saemen. As a consequence, we now have he following heorem. Theorem 3.5. Le he ARFs of order k be defined by (7) and assume ha α k 1 R 0. Furher, suppose ha 1 /αk 1 0 (11) 1 /α j whenever j > k and α j / α k 1, α k, α k. Then i holds ha he form an orhonormal sysem wih respec o a HPDLF M (k). Proof. As poined ou a he end of Secion 2, i suffices o prove ha he L \ L (j 1)\k for j > k. Noe ha L \L (j 1)\k iff K (k) 0. We now have ha K (k 1) j (k 1) 0 for every j > k, due o he fac ha he ARFs are regular. Moreover, as is a HPDLF and because k\(k 1) is regular, we also have ha 1 /αk /α j whenever α j α k 1, α k, α k. Thus, ogeher wih he assumpion given by (11), i follows from Lemma 3.4 ha L \ L (j 1)\k for every j > k. Consequenly, he ARFs hen saisfy he six condiions given in Theorem 2.2. Finally, in Theorem 3.7 we give a relaion beween ARFs of differen order ha holds independen of wheher he ARFs involved form an orhonormal sysem wih respec o a HPDLF. Firs we need he following lemma. Lemma 3.6. The ARFs ϕ (s) n\s κ(s) 0 ˆϕ (s) n\s, wih s k, k + 1, k + 2 and n k + 1, saisfy he relaion given by n\k (x) Z k+1(x) E k+1 + F k+1 ˆϕ (k+1) n\(k+1) (x) Z k+2 (x) + C k+2 Z k (x) ˆϕ(k+2) n\(k+2) (x). (12) Proof. Firs, consider he case in which n k + 1. From Theorem 3.2 we deduce ha (k+1)\k (x) Z k+1(x) E k+1 + F k+1 k\k (x). We also have ha k\k (x) 1 ˆϕ(k+1) (k+1)\(k+1)(x), while (k 1)\k(x) 0 ˆϕ(k+2) (k+1)\(k+2)(x). Hence, he saemen clearly holds for n k + 1.

6 Proceedings of he World Congress on Engineering 2008 Vol II Nex, consider he case in which n k+2. From Theorem 3.2 we now deduce ha (k+2)\k (x) Z k+2(x) E k+2 + F k+2 Z k+1 (x) (k+1)\k (x) Z k+2 (x) + C k+2 Z k k\k (x). Furhermore, we have ha k\k While, Z k+2 (x) E k+2 + F k+2 Z k+1 (x) Z k+2 (x) (x) 1 ˆϕ(k+2) (k+2)\(k+2) (x). (k+1)\k (x) E k+1 + F k+1 Z k+1 (x) E k+2 + F k+2 Z k+1 (x) Z k+1 (x) E k+1 + F k+1 ˆϕ (k+1) (k+1)\(k+1) (x) ˆϕ (k+1) (k+2)\(k+1) (x). Consequenly, he saemen clearly holds for n k + 2 as well. Finally, assume ha he saemen holds for n 2 and n 1. By inducion, he saemen is hen easily verified for n k + 3 by applying he hree-erm recurrence relaion, given by Theorem 3.2, o he lef hand side of (12) for n\k o he righ hand side of (12) for ˆϕ (k+1) n\(k+1), as well as and ˆϕ(k+2) n\(k+2). Theorem 3.7. The ARFs ϕ (s) n\s κ(s) 0 ˆϕ (s) n\s, wih s k, j + 1, j + 2 and k + 1 j n 1, are relaed by n\k (x) ˆϕ(j) n\j (x) Z j+1 (x) + C j+1 Z j 1 (x) ˆϕ(j+1) n\(j+1) (j 1)\k(x). (13) Proof. Noe ha for every l 0 i holds ha ˆϕ (l) l\l (x) 1 and ˆϕ(l) (l+1)\l (x) Z l+1(x) E l+1 + F l+1 Z l (x) Thus, for j n 1 or j k + 1, he relaion given by (13) is nohing more han he hree-erm recurrence relaion given by Theorem 3.2, respecively he relaion given by (12). So, suppose ha he saemen holds for j. By inducion we hen find for j + 1 ha and (j+1)\k(x) ˆϕ(j) (j+1)\j (x) Z j+2 (x) C j+2 Z j (x) ˆϕ(j+2) n\(j+2) (x) ˆϕ(j) n\j (x) + C j+1 Z j+1 (x) Z j 1 (j 1)\k (x), ˆϕ (j+1) n\(j+1) (x) ˆϕ(j) (j+1)\j (x).. Consequenly, ˆϕ (j+1) n\(j+1) (j+1)\k (x) Z j+2 (x) + C j+2 Z j (x) ˆϕ(j+2) n\(j+2) (x) Z j+1 (x) C j+1 Z j 1 (x) ˆϕ(j+1) n\(j+1) (j 1)\k (x) which ends he proof. 4 Conclusion + ˆϕ (j) n\j (x), In his paper, we have given a relaion beween associaed raional funcions (ARFs) of order k 1 and k in erms of raional funcions of he second kind, assuming he ARFs of order k 1 form an orhonormal sysem wih respec o a Hermiian posiive-definie inner produc. Furher, we have given a relaion beween ARFs of differen order ha holds in general; i.e. he relaion holds independenly of wheher he ARFs involved form an orhonormal sysem wih respec o a Hermiian posiive-definie inner produc. If all he poles are a infiniy, we again obain he polynomial case. References 1 A. Bulheel, P. González-Vera, E. Hendriksen, and O. Njåsad, Orhogonal Raional Funcions, Vol. 5 of Cambridge Monographs on Applied and Compuaional Mahemaics, Cambridge Universiy Press, Cambridge, (407 pages). 2 A. Bulheel, P. González-Vera, E. Hendriksen, and O. Njåsad, Orhogonal raional funcions on he real half line wih poles in, 0, J. Compu. Appl. Mah., vol. 179, no. 1-2, pp , July K. Deckers and A. Bulheel, Recurrence and asympoics for orhonormal raional funcions on an inerval, IMA J. Numer. Anal., (Acceped) 4 K. Deckers and A. Bulheel, Orhogonal raional funcions wih complex poles: The Favard heorem, Technical Repor TW518, Deparmen of Compuer Science, Kaholieke Universiei Leuven, Heverlee, Belgium, February J. Van Deun and A. Bulheel, Orhogonal raional funcions on an inerval, Technical Repor TW322, Deparmen of Compuer Science, Kaholieke Universiei Leuven, Heverlee, Belgium, March J. Van Deun and A. Bulheel, Compuing orhogonal raional funcions on a halfline, Rend. Circ. Ma. Palermo (2) Suppl., vol. 76, pp , W. Van Assche, Orhogonal polynomials, associaed polynomials and funcions of he second kind, J. Compu. Appl. Mah., vol. 37, no. 1-3, pp , 1991.