Comlex orthogonal olynomials with the Hermite weight 65 This inner roduct is not Hermitian, but the corresonding (monic) orthogonal olynomials f k g e

Transcription

1 Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 6 (995), 64{74. COMPLEX ORTHOGONAL POLYNOMIALS WITH THE HERMITE WEIGHT Gradimir V. Milovanovic, Predrag M. Rajkovic Dedicated to the memory of Professor Dragoslav S. Mitrinovic In this aer we connect comlex orthogonal olynomials of the Gegenbauer tye on a semicircle with the orthogonal olynomials of the Hermite tye Xn(z). For the last of them, we give the three-term recurrence relation and their relationshi with the classical Hermite olynomials. Also, we study a zero distribution of such olynomials and obtain a linear second-order dierential equation for Xn(z). Some alications in numerical integration are included.. INTRODUCTION In 983 during a joint visit to Henri Poincare Institute at Paris, the rst author of this aer announced Professor Mitrinovic an idea on orthogonal olynomials on the semicircle with a non-hermitian comlex inner roduct. Since he liked this idea, he was always interested about rogress in that direction. Furthermore, he asked Milovanovic to reare a survey about that (see [0]) for his and Keckic's monograh \The Cauchy Method of Residues, Vol. 2 { Theory and Alications" ublished by Kluwer in 993 (and reviously in Serbian by Naucna Knjiga, Belgrade 99). Such olynomials orthogonal on the semicircle = fz 2 C j z = e i ; 0 g have been introduced by Gautschi and Milovanovic [3{4]. The inner roduct is given by (f; g) = f(z)g(z)(iz) dz = f(e i )g(e i )d: 0 0 This work was suorted in art by the Serbian Scientic Foundation, grant number 040F 99 Mathematics Subject Classication: Primary 33A65; Secondary 65D32 64

2 Comlex orthogonal olynomials with the Hermite weight 65 This inner roduct is not Hermitian, but the corresonding (monic) orthogonal olynomials f k g exist uniquely and satisfy a three-term recurrence relation of the form k+ (z) =(z i k ) k (z) k k (z); k =0;;2;:::; (z)=0; 0 (z)=: Notice that the inner roduct ossesses the roerty (zf; g) =(f; zg). The general case of comlex olynomials orthogonal with resect to a comlex weight function was considered by Gautschi, Landau and Milovanovic [2]. Namely, let w : ( ;) 7! R + be a weight function which can be extended to a function w(z) holomorhic in the half disc D + = fz 2 C jjzj<; Im z>0g, and (f; g) = f(z)g(z)w(z)(iz) dz = f(e i )g(e i )w(e i )d: 0 We call a system of comlex olynomials f k g orthogonal on the semicircle if ( k ; m )=0 for k 6= m and ( k ; m ) >0 for k = m (k; m 2 N 0 ); where we assume that k is monic of degree k. The existence of the orthogonal olynomials f k g can be established assuming only that Re (; ) = Re 0 w(e i )d 6=0: Some alications of such olynomials, esecially with the Gegenbauer weight, were given in [9] (see also [8] and [{4). In this aer we consider an orthogonality on a growing semicircle ( ) with radius >, and esecially a limit case when tends to innity (Sections 2 and 3). In Section 4 we discuss alications in numerical integration. 2. COMPLEX POLYNOMIALS ORTHOGONAL ON A GROWING SEMICIRCLE Let w (z) be the Gegenbauer weight function, w (z) =( z 2 ) =2 (>0); and D + the half disc D + = z 2 C jjzj; Im z>0, bounded by the semicircle and the interval [ ; ]. In [2] itwas dened an inner roduct on the semicircle with resect to the weight function w(z) which is not Hermitian. Namely, (f; g) = f(z)g(z)w (z)(iz) dz:

3 66 Gradimir V. Milovanovic, Predrag M. Rajkovic But, it was roved that there exists an unique sequence of monic olynomials n (z) such that is ( m ; n)= mn k nk 2 (m; n =0;;:::): A connection with the monic Gegenbauer olynomials b C n (z) was also found, n (z) = b C n (z) i n b C n (z); where = ; 0 = ( +=2) ; ( +) k = +k ((k +2)=2) ( +(k+)=2) : ((k +)=2) ( + k=2) The norm of such olynomials is given by k n k2 = ( 0 n )2 >0 (n): Introduce now a new variable u by u = z. Then the semicircle becomes a new one (see Fig. 2.) ( ) = u 2 C jjuj ; Im u>0 : The orthogonality condition becomes O 6 Fig. 2. () m u= n u= u 2 = =2 du iu = mnk nk 2 : ( ) - We can dene a new sequence of olynomials X n (u) by X n(u) = n n u= ; which are orthogonal with resect to the weight u ew (u) = 2 on ( ).Thus, (2:) hx m ;X ni = =2 ( ) X m(u)x n(u)ew (u) du iu = mnkx nk 2 : Putting P n (t) = nb C n(t= )(n2n 0 ), we see that this sequence satises the following three-term recurrence relation P k+ (t) =tp k (t) b k P k (t); P (t) =0; P 0 (t)=;

4 where Comlex orthogonal olynomials with the Hermite weight 67 + b 0 = t 2 =2 dt = ( +=2) ( +) and b k(k +2 ) k = 4(k + )(k + ) ; k : These olynomials are orthogonal on ( ; ) with resect to ew (t). Then, we yield (2:2) X n(u) =P n(u) i n P n (u); where = ; n = n (n ): Similarlyas in[2{3]we can rove that they satisfy three-term recurrence relation and a second order dierential equation. Their zeros lie in the region bounded by ( ) and [ ; ]. Using Cauchy's theorem the inner roduct h ; i can be exressed in the form (2:3) hf; gi = f(0)g(0) + i v.. + f(x)g(x) w (x) x dx: 3. COMPLEX POLYNOMIALS ORTHOGONAL WITH THE HERMITE WEIGHT It is known (see Szeg}o [7,. 07]) that is (3:) lim!+ P k (t) = b Hk (t): where b Hk (t), k =0;;:::, are the monic Hermite olynomials which satisfy bh k+ (t) =tb Hk (t) b k b Hk (t); b H (t)=0; b H0 (t)=; with b 0 = and b k = k=2 (k). Dening k = lim!+ () k,we nd Knowing that (cf. [6]) = ; n = ((n +2)=2) ((n +)=2) (n =0;;:::): lim x!+ (x + a) x a (x) = (a>0);

5 68 Gradimir V. Milovanovic, Predrag M. Rajkovic we conclude that lim n!+ n = lim (n +)=2=+. n!+ Now, we can dene a sequence of comlex olynomials X n (z) by X n (z) = lim!+ X n(z): Using (3:) and (2:2) we conclude that these olynomials satisfy the following recurrence relation X n (z) = b Hn (z) i n b Hn (z); X (z) =0; X 0 (z)=: Recently, Notaris [5, Lemma 2.] roved that lim!+ + q () k (t) ew (t)dt= + q k (t)e t2 dt for any monic olynomial q m () of degree m, whose coecients deend on a arameter, such that lim m (t) =q m (t), where q m is a monic olynomial of degree!+ q() m. Therefore, from (2:) and (2:3) we obtain the inner roduct (3:2) hf; gi = f(0)g(0) + i v.. + f(x)g(x) e x2 x dx: Theorem 3. The sequence ofolynomials fx n (z)g is orthogonal with resect the inner roduct (3:2), i.e., hx m ;X n i= mn kx n k 2 (m; n =0;;:::): Remark 3. The sequence fx n (z)g can be introduced using two functionals (see P. Maroni [7]): L for a real olynomial sequence and u for the corresonding comlex olynomial sequence. If L is the Hermite functional, then the functional u = c +(x c) L generates the sequence fx n (z)g. Theorem 3.2. The sequence fx n (z)g satises the three-term recurrence relation where X n+ (z) =(z i n )X n (z) n X n (z); X (z) =0; X 0 (z)=; 0 = 0 ; 0 = ; n = n n ; n = 2 n (n ): The norm of olynomials is given by kx n k 2 = hx n ;X n i= 0 n = n+ 2 2 (n0):

6 Comlex orthogonal olynomials with the Hermite weight 69 Examle 3. A few values of n and X n (z) are given by 0 = ; X 0 (z) =; = 2 ; X (z)=z i ; 2 = 2 ; X 2 (z)=z 2 i 2 z 2 ; 3 = 3 ; X 3 (z)=z 3 i 2 z z+i ; 4 = 8 3 ; X 4(z)=z 4 i 3 4 z3 3z 2 +i 9 8 z+3 4 : Notice that =. Remark 3.2 Introducing hf; gi = f(0)g(0) + i v.. + f(x)g(x)jxj s x 2 =2 dx; when! +, we obtain the inner roduct whose corresonding orthogonal olynomials are known as the generalized Hermite olynomials. Since zeros of the monic Hermite olynomials b Hn (z) (n=;2;:::) are real, simle and satisfy the searation theorem, for the olynomials X n (z) wehave: Theorem 3.3. All zeros of X n (z) are contained in the rectangle R + = fz 2 C j n Re z n ; 0 < Im z< n =2g; where n is the largest zero of b Hn (z). Proof. At rst, we note that X n (z) cannot have any real zero. suose that exists a real such that X n () = 0, i.e., Indeed, if we bh n () i n b Hn () =0; then it must be b Hn () = b Hn () = 0, because b Hn (), b Hn (), and n are real. However, this is a contradiction with the searation theorem. According to a result of Giroux [5, Corollary 3] (see also [2,. 269]) all zeros of X n (z) either lie in the half stri S + = fz 2 C j n Re z n ; 0 < Im zg ; or in the conjugate half stri. Since X n (z) =z n i n z n, using the Viete rule, we nd nx j= j = i n ;

7 70 Gradimir V. Milovanovic, Predrag M. Rajkovic from which we conclude that Im n n X j= j o = n > 0; i.e., Im j > 0, j =;:::;n. Like in[2], we can rove that all zeros of X n (z) are located symmetrically with resect to imaginary axis because of the symmetric weight. It also gives bounds for zeros. If n is odd, then X n (z) has one urely imaginary zero. 2 Remark 3.3. An interesting result on the zero distribution for olynomials orthogonal on the semicircle can be found in []. Theorem 3.4. The olynomial X n (z) satises the dierential equation (3:3) P (z)x 00 n(z) 2 zp(z)+i n X 0 n (z)+2 np (z) 2 2 n Xn (z) =0; where P (z) =2i n z +2 2 n n: Proof. We can rove it starting with the function which satises (z) =e z2 +2i n z ; (z) b Hn (z) 0 = A (z)xn (z); A = const, by the same rocedure as in [2]. A general way for nding such dierential equations was given in [7]. 2 Dividing (3.3) by P (z) we obtain the equation X 00 n(z) 2 z + i n P (z) X 0 n(z)+2 n 2 2 n P(z) X n (z)=0; which is more similar to the Hermite equation. For n = 0, the olynomial X n (z) becomes the olynomial Hn b (z) and the dierential equation is the corresonding one. 4. QUADRATURES OF GAUSSIAN TYPE In this section we construct a Gaussian quadrature formula (4:) L(f) = for the functional nx = (4:2) L(f) = f(0) + i v.. f( )+R n (f); R n (P 2n )=0; + t f(t) e t2 dt:

8 Comlex orthogonal olynomials with the Hermite weight 7 Here P m denotes the set of all olynomials of degree at most m. Taking X n (z) = ny k= we obtain an interolatory quadrature (4.) if (z k ); = Xn( 0 ) L X n()=( ) ; =;:::;n: This formula will be of Gaussian tye if and only if the node olynomial X n is chosen in such away to be orthogonal to P n with resect to the functional L, i.e., to the inner roduct (3.2). Thus, X n must be given as in Theorem 3.. It is easy to nd the nodes and weights in an analytic form when n = and n = 2. Namely, for n =wehave =i= and =, and for n = 2 the arameters are 8 ;2 = + i 0: : i 4 4 and ;2 = 2 i(4 ) 2 : : i: 8 For f(z), from (4.) and (4.2) we obtain that and nx = = : Letting e Xk (z) =X k (z)=kx k kdenote the normalized orthogonal olynomials X(z) = e X0 (z);e X (z);:::; e Xn the vector of the rst of them, it is easily seen that (z) T z e Xk (z) = k e Xk (z)+i k e Xk (z)+ k e Xk+ (z); k =0;;:::; and J n X( )= X( ); where J n = i 0 0 O 0 i i n : O n 2 i n

9 72 Gradimir V. Milovanovic, Predrag M. Rajkovic The nodes are therefore the eigenvalues of the Jacobi matrix J n and X( ) the corresonding eigenvalues. By an adatation of the rocedure of Golub and Welsch [6] asin[4] and [9] and using the EISPACK routine HQR2 and the LINPACK routines CGECO and CGESL we can comute the arameters of the Gaussian quadrature (4.). In Table 4. we dislay these arameters (to 8 decimals only, to save sace) for n = 5, 0, 20 (numbers in arentheses denote decimal exonents). Table 4.. Gaussian formula for n =5;0; 20 n 5,2 : : i 0: ( ) 0:53438( ) i 3,4 0: :32244 i 0: : i 5 0: i 2: ,2 3: : i 0: ( 5) 0:296059( 5) i 3,4 2: : i 0: ( 3) 0: ( 3) i 5,6 : : i 0: ( ) 0: ( ) i 7,8 0: : i 0: : i 9,0 0: : i : : i 20,2 5: : i 0: ( 3) 0: ( 3) i 3,4 4: : i 0: ( 9) 0:303066( 9) i 5,6 3: : i 0: ( 7) 0: ( 7) i 7,8 3: : i 0: ( 5) 0: ( 5) i 9,0 2: : i 0:086797( 3) 0:033727( 3) i,2 2: : i 0: ( 2) 0: ( 2) i 3,4 : : i 0: ( ) 0: ( ) i 5,6 : :98448 i 0:768493( ) 0:968543( ) i 7,8 0: : i 0: : i 9,20 0: : i : : i We notice that is real if is urely imaginary; and that is + = if + =. An interesting alication of Gaussian formulae (4.) could be to Cauchy rincial value integrals. Let z 7! f(z) be a holomorhic function in Im z 0. Then we have v.. + f(t) e t2 dt i t In articular, if f(z) is real for real z, then (4:3) v.. + n f(0) f(t) e t2 dt Im t nx = nx = f( ) f( ): Examle 4.. We aly (4.3) to Cauchy rincial value integral I = v.. + e t t e t2 dt =: ::: : o :

10 Comlex orthogonal olynomials with the Hermite weight 73 Table 4.2. Gaussian aroximation of Cauchy rincial value integral I and relative errors n Aroximation Rel. error 2 : :2( 3) 3 : :70( 4) 4 :9394 7:70( 6) 5 : :78( 7) 6 : :47( 9) 7 : :84( ) 8 : :65( 3) 9 : :3( 4) 0 : :5( 6) The obtained results for n = 2()0, with relative errors, are given in Table 4.2. In each entry the rst digit in error is underlined. REFERENCES. W. Gautschi: On the zeros of olynomials orthogonal on the semicircle. SIAM J. Math. Anal. 20 (989), 738{ W. Gautschi, H. J. Landau, G. V. Milovanovic: Polynomials orthogonal on the semicircle, II. Constr. Arox. 3 (987), 389{ W. Gautschi, G. V. Milovanovic: Polynomials orthogonal on the semicircle. Rend. Sem. Mat. Univ. Politec. Torino (Secial Functions: Theory and Comutation), 985, 79{ W. Gautschi, G. V. Milovanovic: Polynomials orthogonal on the semicircle. J. Arox. Theory 46 (986), 230{ A. Giroux: Estimates for the imaginary arts of the zeros of a olynomials. Proc. Amer. Math. Soc. 44 (974), 6{ G. H. Golub, J. H. Welsch: Calculation of Gauss quadrature rules. Math. Com. 23 (969), 22{ P. Maroni: Sur la suite de olyn^omes orthogonaux associee a la forme u = c + (x c) L: Period. Math. Hung. 2 (990), 223{ G. V. Milovanovic: Some alications of the olynomials orthogonal on the semicircle. In: Numerical Methods (Miskolc, 986), Colloq. Math. Soc. Janos Bolyai, Vol. 50, North-Holland, Amsterdam { New York 987, 625{ G. V. Milovanovic: Comlex orthogonality on the semicircle with resect to Gegenbauer weight: Theory and alications. In: Toics in Mathematical Analysis (Th.M. Rassias, ed.), World Scientic Publ., Singaore 989, 695{722.

11 74 Gradimir V. Milovanovic, Predrag M. Rajkovic 0. G. V. Milovanovic: Comlex olynomials orthogonal on the semicircle. In: The Cauchy Method of Residues, Vol. 2 { Theory and Alications (D. S. Mitrinovic and J. D. Keckic), Kluwer, Dordrecht 993, 47{6.. G. V. Milovanovic: On olynomials orthogonal on the semicircle and alications. J. Comut. Al. Math. 49 (993), 93{ G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias: Toics in Polynomials: Extremal Problems, Inequalities, eros. World Scientic, Singaore { New Jersey { London { Hong Kong G. V. Milovanovic, P. M. Rajkovic: Geronimus concet of orthogonality for olynomials orthogonal on a circular arc. Rend. di Matematica, Serie VII (Roma) 0 (990), 383{ G. V. Milovanovic, P. M. Rajkovic: On olynomials orthogonal on a circular arc. J. Comut. Al. Math. 5 (994), {3. 5. S. E. Notaris: Some new formulae for the Stieltjes olynomials relative to classical weight functions. SIAM J. Numer. Anal. 27 (99), 96{ J. G. Wendel: Note on the gamma function. Amer. Math. Monthly 55 (948), 563{ G. Szeg}o: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, R. I Deartment of Mathematics, (Received Setember, 995) Faculty of Electronic Engineering, University ofnis, P.O. Box 73, 8000 Nis, Yugoslavia Deartment of Mathematics, Faculty of Mechanical Engineering, University ofnis, Beogradska 4, 8000 Nis, Yugoslavia