37 Kragujevac J. Math. 23 (2001) A NOTE ON DENSITY OF THE ZEROS OF ff-orthogonal POLYNOMIALS Gradimir V. Milovanović a and Miodrag M. Spalević

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1 37 Krgujevc J. Mth. 23 (2001) A NOTE ON DENSITY OF THE ZEROS OF ff-orthogonal POLYNOMIALS Grdimir V. Milovnović nd Miodrg M. Splević b Fculty of Electronic Engineering, Deprtment of Mthemtics, University of Ni»s, P. O. Box 73, Ni»s, Serbi, Yugoslvi b Fculty of Science, Deprtment of Mthemtics nd Informtics, P. O. Box 60, Krgujevc, Serbi, Yugoslvi (Received My 15, 2001) Abstrct. A new proof of density of the zeros of ff-orthogonl polynomils is presented. Some numericl results for clss of ff-orthogonl polynomils on two symmetric intervls re included. INTRODUCTION Let d (t) be nonnegtive distribution on the compct support [; b], for which ll moments μ k = t k d (t); k =0; 1;:::; exist nd re finite, nd μ 0 > 0. Let ff = (s 1 ;s 2 ;:::;s n ;:::) be bounded sequence of nonnegtive integers, nd denote (s 1 ;s 2 ;:::;s n ) by ff n. Let s = mxfs k j k =1; 2;:::g:

2 38 Assume tht for fi (= fi (ffn) ) 2 (; b), =1;:::;n, hold fi 1 <fi 2 < <fi n : It is known (for more detils see the survey pper [5]) tht the Chklov-Popoviciu qudrture formul with multiple nodes, f(t) d (t) = hs the mximum degree of exctness if nd only if =1 d mx =2 nx 2s X =1 h=0 ψ n X =1 A h f (h) (fi )+R(f); (1)! s + n 1 (t fi ) 2s+1 t k d (t) =0; k =0; 1;:::;n 1: (2) The proofs of the existence nd the uniqueness of (1) which hve been obtined recently cn be found in [6], [7]. The orthogonlity conditions (2) define sequence of polynomils fß n;ff g n2n 0 which re clled ff-orthogonl polynomils. In the cse s 1 = s 2 = ::: = s the bove polynomils reduce to the s-orthogonl polynomils, nd (1) is known s the Guss-Turán qudrture formul. MAIN RESULT Recently Shi [8] hs stted property of density offf-orthogonl polynomils. He obtined this sttement s consequence of the convergence of the corresponding qudrture formul (1) for f 2 C 2s [; b]. The min result of our pper cn be stted in the following form: Theorem. Let d (t) be nonnegtive distribution on the finite segment [; b], nd let fß n;ff (t)g denote the ssocited orthogonl set of (monic) ff-polynomils. Let

3 39 [ 0 ;b 0 ] be subintervl of [; b] such tht R b 0 every polynomil ß n;ff (t) hs t lst one zero in [ 0 ;b 0 ]. Proof. 0 d (t) > 0. Then if n is sufficiently lrge, Let %(t) be n rbitrry polynomil of degree m, which is not greter thn 0in[; b], except possibly in [ 0 ;b 0 ]. Assuming tht the polynomil ß k;ff (t) (k = n; n +1;:::) hs no zeros fi (ff k) in [ 0 ;b 0 ], nd tking 2n 1 m (therefore 2k 1 m (k = n; n +1;:::)), we obtin %(t) ky =1 (t fi (ff k) ) 2s d (t) = kx =1 (ff k) %(fi (ff k) ) (» 0); (3) where (3) is the corresponding Guss qudrture formul subject to the new nonnegtive distribution dμ(t) =dμ (ff k) (t) = ky =1 (t fi (ff k) ) 2s d (t): (A generl clss implicitly defined polynomils ws introduced nd studied by Engels (cf. [2, pp ]).) It is cler tht the condition R b 0 0 d (t) > 0 implies R 0 b k) (t) > 0. 0 dμ(ff Hence, when we pply the theorem of Weierstrss, it follows tht f(t) dμ (ff k) (t)» 0; where f(t) is continuous in [; b] nd not greter thn 0 in [; b], except possibly in [ 0 ;b 0 ]. If we define (see Szegö [9, pp ]) ( 0; in» t» 0 nd b f(t) = 0» t» b; (t 0 )(b 0 t); in 0» t» b 0 ; we rech contrdiction. Remrk. In the specil cse we hve tht the bove result holds for the zeros of s-orthogonl polynomils. By different method it ws lso proved by Mrtinelli, Ossicini nd Rosti [4]. Let [ 0 ;b 0 ] ρ [; b], with b 0 0 < b, nd (t) is constnt on [ 0 ;b 0 ] ( ( 0 ) = (b 0 ) = const). We cn prove tht ß n;ff (t) (n 2) hs t most one zero in [ 0 ;b 0 ].

4 40 As we know, the n zeros of ß n;ff (t) re distinct, rel nd ll contined in the open intervl (; b) (see [5]). Let ß n;ff (t) hs t lest two zeros in [ 0 ;b 0 ], nd fi nd T re the miniml nd mximl zero of ß n;ff (t) belong to [ 0 ;b 0 ], respectively. Consider polynomil of degree n 2 defined by for which holds We hve =1 = (t fi ) 2s +1 = =1 ß n;ff (t) (t fi)(t T ) ; (t fi ) 2s+1 d (t) =0: (4) =1 (t fi ) 2s+2 (t fi)(t T ) 0; for t =2 (fi;t); nd =1 (t fi ) 2s+1 d (t) = + Z 0 b 0 =1 =1 (t fi ) 2s +1 d (t) (t fi ) 2s+1 d (t) > 0; nd this contrdicts the condition (4). SOME NUMERICAL RESULTS In this section we consider clss of ff-orthogonl polynomils on two symmetric intervls. The theory of polynomils orthogonl on [ 1; 1] with respect to the mesure d (t) =w(t)dt, where w(x) = 8 < : jt + ffj(t 2 ο 2 ) p (1 t 2 ) q ; t 2 [ 1; ο] [ [ο;1]; 0; elsewhere; nd 0 < ο < 1, p; q > 1, hs been studied by Brkov [1] nd Gutschi [3] (when jt + ffj ws replced by the symmetric fctor jtj fl, fl 2 R). (5)

5 41 Tble 1: Zeros fi (ffn) for n = 2(1)6 nd r = 0:8; 0:4; 0:1, nd 0:01 n r =0:8 r =0:4 r =0:1 r =0:01 ο =0:11111 ο =0:42857 ο =0:81818 ο =0: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The specil cse ff = 0, p = q = 1=2, ο = (1 r)=(1 + r) (0 < r < 1) of (5) rises in the study of the ditomic liner chin, where r = m=m hs the mening of mss rtio, m nd M (m < M) being the msses of the the two kinds of prticles lternting long the chin (see [10] nd [3]). In tht cse, the coefficients in the three-term recurrence reltion for polynomils fß n g, orthogonl in usul sense (s 1 = s 2 = =0)with respect to d (t), ß n+1(t) =tß n (t) fi n ß n 1(t); ß 1 (t) =0; ß 0 (t) =1; re known explicitly fi 0 = ß; fi 1 = 1 2 (1 + ο2 ); fi 2 = 1 4 (1 ο2 ) 2 1+ο 2 ; fi 3 = ο2 + ο 4 1+ο 2 ;

6 42 fi 2k = 1 16 (1 ο2 ) 2 fi 2k 1 ; fi 2k+1 = 1 2 (1 + ο2 ) fi 2k (k =2; 3;:::): Using the itertive procedure for determining the zeros of ff-orthogonl polynomils given in [6], we illustrte the results from the previous section. We tke ff-sequence, for exmple, ff = f2; 1; 3; 4; 0; 2;:::g, nd clculte the zeros of ß n;ff (t) for some selected vlues of r, i.e., ο, nd n. The corresponding zeros fi (ffn) re presented in Tble 1. The boxed zeros belong to the internl intervl [ ο;ο]. Notice tht t most one zero of the polynomil ß n;ff (t) cn be inside this hole," which spreds when r decreses. In the other words, t lest n 1 zeros of of this polynomil is very close to the points ±1, when r! 0. In the cse of s-orthogonl polynomils with respect to the weight (5) (for ff =0), the corresponding zeros re symmetriclly distributed round the origin, so tht only polynomils of odd degree hs one zero in t =0. The ll zeros of polynomils of even degree re outside the hole" [ ο;ο]. For s = mx ff k, n = 2(1)6, nd for the sme 1»k»n selected vlues of r s before, these zeros re given in Tble 2. Tble 2: Zeros of s-orthogonl polynomils for n = 2(1)6 nd r = 0:8; 0:4; 0:1, nd 0:01 n s r =0:8 r =0:4 r =0:1 r =0:01 ο =0:11111 ο =0:42857 ο =0:81818 ο =0: : : : : : : : : : 0: 0: 0: 4 4 0: : : : : : : : : : : : : : : : : 0: 0: 0: 6 4 0: : : : : : : : : : : :

7 43 References [1] G.I. Brkov, On some systems of polynomils orthogonl on two symmetric intervls, Izv. Vys»s. U»cebn. Zv. Mt. 1960, No. 4 (17), [2] H. Engels, Numericl Qudrture nd Cubture, Acdemic Press, London, [3] W. Gutschi, On some orthogonl polynomils of interest in theoreticl chemistry, BIT 24 (1984), [4] M.R. Mrtinelli, A. Ossicini, nd F. Rosti, Densit degli zeri di un sistem di polinomi s-ortogonli, Rend. Mt. 12 (1992), [5] G.V. Milovnović, Qudrtures with multiple nodes, power orthogonlity, nd moment-preserving spline pproximtion, in: W. Gutschi, F. Mrcelln, L. Reichel (Eds.), Numericl nlysis 2000, Vol. V, Qudrture nd orthogonl polynomils. J. Comput. Appl. Mth. 127 (2001), [6] G.V. Milovnović nd M.M. Splević, Qudrture formule connected to fforthogonl polynomils, J. Comput. Appl. Mth. (to pper). [7] Y.G. Shi, Generlized Gussin qudrture formuls for Tchebycheff systems, Fr Est J. Appl. Mth. 3 (1999), [8] Y.G. Shi, Convergence of Gussin qudrture formuls, J. Approx. Theory 105 (2000), [9] G. Szeg}o, Orthogonl Polynomils, Amer. Mth. Soc. Colloq. Publ., Vol 23, 4th ed., Amer. Mth. Soc., Providence R.I., [10] J.C. Wheeler, Modified moments nd continued frction coefficients for the ditomic liner chin, J. Chem. Phys. 80 (1984),