Zero Location for Nonstandard Orthogonal Polynomials

Transcription

1 Journal of Approxiation Theory 113, (2001) doi: /jath , available online at on Zero Location for Nonstandard Orthogonal Polynoials A. J. Duran 1 Departento Análisis Mateático, Facultad de Mateáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla, Spain E-ail: duran@cica.es and E. B. Saff 2 Departent of Matheatics, Vanderbilt University, Nashville, Tennessee E-ail: esaff@ath.vanderbilt.edu Counicated by Walter Van Assche Received Noveber 3, 2000; accepted in revised for April 11, 2001 A ethod to locate the zeros of orthogonal polynoials with respect to nonstandard inner products is discussed and applied to Sobolev orthogonal polynoials and polynoials satisfying higher-order recurrence relations Acadeic Press Key Words: zeros; higher-order recurrence relation; atrix polynoials; Sobolev inner product. 1. INTRODUCTION In this note (Section 2) we develop a ethod for the location of the set of zeros of a sequence of orthogonal polynoials with respect to an inner product of the for (1.1) Op, qp=f (T 0 (p),..., T N (p)) W(z)(T 0 (q),..., T N (q)) g d(z), where: (i) T 0,...,T N are linear operators on the space P of algebraic polynoials with coplex coefficients, and 1 The research of this author was partially supported by DGICYT Ref. PB The research of this author was supported, in part, by U.S. National Science Foundation Grant DMS /01 $35.00 Copyright 2001 by Acadeic Press All rights of reproduction in any for reserved.

2 128 DURAN AND SAFF (ii) W(z) is a positive definite atrix of integrable functions with respect to the positive easure supported on a subset of the coplex plane. The ethod is applied to locate zeros of Sobolev orthogonal polynoials and polynoials satisfying higher-order recurrence relations (assuing that the atrix of functions W is diagonal). In both cases the bounds for the zeros are sharp. In Section 3, we consider orthogonal polynoials with respect to a Sobolev inner product of the for (1.2) N Op, qp= C F p (k) (z) q (k) (z) w k (z) d(z), N \ 1, k=0 where is a finite positive Borel easure with support D C (D containing infinitely any points) and w k L 1 (), w k \ 0, k=0,..., N. These nonstandard inner products have attracted uch attention in the last few years. Regarding the zero location, soe results have been obtained for the case when has real support and w k (z) d(z) reduces to a Dirac s delta, k=1,..., n, (discrete Sobolev inner product); see for instance ([AMRR1], [AMRR2]). For a continuous Sobolev inner product, W. Gautschi and A. B. J. Kuijlaars in [GK] have found zero asyptotic properties for N=1 under the hypothesis that w 0 d and w 1 d are regular easures (in the sense of [ST]). More recently, G. Lopez and H. Pijeira in [LP] have obtained soe estiates on the zeros of Sobolev orthogonal polynoials with respect to an inner product of the for given in (1.2) assuing that is supported on a copact set of the real line and that w k /w k 1 L. (), k=1,..., N. They used the boundedness of the ultiplication operator defined in a certain Banach space associated with the Sobolev inner product. G. Lopez has pointed out to the authors a new and very short proof for the boundedness of the zeros assuing the boundedness of the ultiplication operator which, for the sake of copleteness, we include here (see also [LPP]): let (p n ) n be a sequence of orthogonal polynoials with respect to an inner product O, P for which the ultiplication operator is bounded, that is, there exists c>0 such that Ozq(z), zq(z)p [ coq(z), q(z)p for any polynoial q; then a [ `c for any zero a of p n, n N. Indeed, we can write p n (z)=(z a) q(z) for soe polynoial q of degree n 1; then Op n (z), q(z)p=0 and since zq(z)=p n (z)+aq(z) we get that Ozq(z), zq(z)p=op n (z), p n (z)p+oaq(z), aq(z)p \ a 2 Oq(z), q(z)p.

3 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 129 Using the boundedness of the ultiplication operator we have that a 2 Oq(z), q(z)p [ Ozq(z), zq(z)p [ coq(z), q(z)p, and since Oq(z), q(z)p ] 0, we finally deduce that a [ `c. Let us reark that, for Sobolev inner product, the boundness of the ultiplication operator forces the easures which define the inner product to have copact support. Using our ethod, we can extend the latter results (iproving also their bounds) for easures with support in any set of the coplex plane (copact or not): Theore 1.1. Consider a Sobolev inner product of the for given in (1.2), where is a finite Borel easure with support D in the coplex plane C (D containing infinitely any points) and w k L 1 (), w k \ 0, k=0,..., N. Assue w k /w k 1 L. (), k=1,..., N, and write C k = w k /w k 1., k= 1,..., N. Ifz 0 is a zero of an orthogonal polynoial with respect to O,P, then (1.3) d(z 0, Co(D)) [ (1/2)= C N k=1 k 2 C k, where Co(D) is the convex hull of the support of. An exaple is presented in Section 3 showing that the estiate (1.3) is sharp. We coplete Section 3 obtaining other different bounds (see Theore 3.2) and extending the for general Sobolev inner products of size 2 2 (see Theore 3.3). In Section 4 we consider sequences of polynoials (p n ) n satisfying a (2N+3)-ter recurrence relation of the for (1.4) N+1 t N+1 p n (t)=c n, 0 p n (t)+ C [c n, k p n k (t)+c n+k, k p n+k (t)], k=1 with initial conditions p k =0, for k<0, and p k (t) a polynoial of degree k, for k=0,..., N+1. One of the authors proved in [D2] that the sequence (p n ) n is then orthonoral with respect to an inner product of the for (1.1), where (1.5) T (p)=c n p (n(n+1)+) (0) (n(n+1)+)! tn,

4 130 DURAN AND SAFF and a positive easure supported on the real line. Discrete Sobolev orthonoral polynoials are a particular case of polynoials satisfying a higher order recurrence relation (see [DV], Sect. 3). Polynoials satisfying higher-order recurrence relations such as (1.4) are closely related to orthogonal atrix polynoials (see [D1], [D2], [DV]). We next assue that the atrix of functions W is diagonal and has copact support. Using our ethod we get the following bound: Theore 1.2. Consider an inner product of the for (1.6) N Op, qp= C F T k (p(z)) T k (q(z)) w k (z) d(z), k=0 where the operators T, =0,..., N, are given by (1.5), is a finite positive Borel easure with copact support D C (D containing infinitely any points) and w k L 1 (), w k \ 0. Assue that w k /w k 1 L. (), k=1,..., N, and z 2 w 0 (z)/w N (z) L. (). Ifz 0 is a zero of an orthogonal polynoial with respect to O,P, then (1.7) z 0 [ ax{`c 0, `C 1,...,`C N }, where C k = w k /w k 1., k=1,..., N, and C 0 = z 2 w 0 (z)/w N (z).. We provide an exaple that shows (1.7) is sharp. Finally, we extend Theore 1.2 for the nondiagonal case of size 2 2 (see Theore 4.1). 2. DESCRIPTION OF THE METHOD We consider inner products O,P, defined on the linear space of polynoials with coplex coefficients, of the for given by (1.1). We will assue that for each z in the support of and fixed w C there exists a (N+1) (N+1) atrix C(z, w) such that for all p P (2.1) (T 0 ((z w) p(z)),..., T N ((z w) p(z))) =(T 0 (p(z)),..., T N (p(z))) C(z, w). Let us reark that for an inner product these atrices C do not exist in general (e.g. consider T 0 (p)=> 1 0 p(t) dt and T 1 (p)=pœ(0); then for p(t)= 1 3t 2 one has T 0 (p)=t 1 (p)=0, but T 0 ((z w) p)= 1/4, T 1 ((z w) p)=1). However for the ost interesting cases of inner products of the for (1.2) and (1.6) such atrices C always exist (see (3.1) for inner products of the for (1.2) and (4.1) for those of the for (1.6)).

5 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 131 Let z 0 be a zero of an orthogonal polynoial p for the inner product O,P. Write p(z)=(z z 0 ) q(z). Then fro the orthogonality of p and (2.1) we get for any a C that 0=aO(z z 0 ) q(z), q(z)p+āoq(z), (z z 0 ) q(z)p =a F [(T 0 ((z z 0 ) q(z)),..., T N ((z z 0 ) q(z))) W(z)(T 0 (q(z)),..., T N (q(z))) g +ā(t 0 (q(z)),..., T N (q(z))) W(z)(T 0 ((z z 0 ) q(z)),..., T N ((z z 0 ) q(z))) g ]d =F (T 0 (q(z)),..., T N (q(z)))(ac(z, z 0 ) W(z)+āW(z) C(z, z 0 ) g ) (T 0 (q(z)),..., T N (q(z))) g d. We can then conclude that for soe z in the support of the heritian atrix (2.2) (ac(z, z 0 ) W(z)+āW(z) C(z, z 0 ) g ) cannot be positive definite or negative definite. Fro this fact one can obtain soe bounds on z 0 under additional hypotheses on the inner product. In the next sections we illustrate the ethod with soe exaples. 3. SOBOLEV INNER PRODUCTS In this section we prove the estiate (1.3) stated in Theore 1.1 (we thus assue that the atrix of functions W is diagonal). Indeed, for a Sobolev inner product we have that T =D () and then fro which we deduce: ((z w) p(z)) () =(z w) p () (z)+p ( 1) (z), (3.1) z w z w C(z, w)=rz w x x x z z x z w N z w S.

6 132 DURAN AND SAFF Let z 0 be a zero of an orthogonal polynoial with respect to the Sobolev inner product given by (1.2). If z 0 Co(D) there exists z 1 Co(D) such that d(z 0, Co(D))=d(z 0,z 1 )>0. We first assue that Iz 0 =Iz 1 and that Rz 1 \ Rz 0. This iplies that Rz Rz 0 > Rz 1 Rz 0 >0for z Co(D) and that (3.2) d(z 0, Co(D))=inf{Rz Rz 0 :z Co(D)}. Taking a=1 in (2.2) and according to our ethod we have that the heritian atrix C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g cannot be positive definite for soe z D. Forula (3.1) now gives that: (3.3) C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g z0)w0(z) w1(z) w 1 (z) 2R(z z 0 )w 1 (z) 2w 2 (z) =R2R(z 0 2w 2 (z) 2R(z z 0 )w 2 (z) 3w 3 (z)... 0 S. x x z z z x Nw N (z) 2R(z z 0 )w N (z) To prove (1.3) we need the following bound on the principal deterinant of this atrix. Lea 3.1. Write M n =M n (z, z 0 ) for the nth principal deterinant of the atrix C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g.ifm n >0 for 1 [ n [, where 0 [ [ N, then M +1 \ (2(Rz Rz 0 )) 1 1 4(Rz Rz0 ) 2 D k=0 w k (z) C k=1 k 2 w 2 k(z) D i=0; i ] k, k 1 w i (z)2. Proof. Let a=rz Rz 0. Expanding M n+1 by the last row we get the following recurrence forula: (3.4) M n+1 =2aw n M n n 2 w 2 nm n 1. Since M n >0for 1 [ n [, it follows that (3.5) M n+1 [ 2aw n M n.

7 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 133 This gives for n=: M +1 =2aw M 2 w 2 M 1 \ 2aw M 2a 2 w 2 w 2 M 2. Using the recurrence relation (3.4) for n= 1 we get: M +1 \ (2a) 2 w w 1 M 1 Again fro (3.5) for n= 3 we get (2a)( 2 w 2 w 2 +( 1) 2 w w 2 1)M 2. M +1 \ (2a) 2 w w 1 M 1 (2a) 2 ( 2 w 2 w 2 w 3 +( 1) 2 w w 2 1w 3 )M 3. Applying successively (3.4) and (3.5) we find that M +1 \ (2a) 2 w w 3 M 3 (2a) 2 1 C k=3 k 2 w 2 k Lea 3.1 follows now by taking into account that M 3 =(2a) 3 w 2 w 1 w 0 (2a)(2 2 w 2 2w 0 +w 2 1w 2 ), D w i 2 M1. i=1; i ] k, k 1 and M 1 =2aw 0. L Since the heritian atrix C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g cannot be positive definite, we have that for soe +1, 1 [ [ N, and for soe z in D, M +1 (z, z 0 ) [ 0. We take the sallest with this property. Fro Lea 3.1 and taking into account that Rz Rz 0 >0 for z D, we deduce that 4(Rz Rz 0 ) 2 D k=0 w k (z) C k=1 k 2 w 2 k(z) D i=0; i ] k, k 1 w i (z) [ 0. This inequality finally gives for soe z D and soe, 1 [ [ N, that = Rz Rz 0 [ 1 C 2 k=1 k 2 w k (z) w k 1 (z). To get (1.3), it is enough to take into account (3.2) and that the right-hand side in the previous inequality is an increasing function of.

8 134 DURAN AND SAFF If Iz 0 ] Iz 1,orRz 1 < Rz 0, we proceed as follows. We write r for the line joining z 0 and z 1 and s for the line perpendicular to r at z 1. We now ake a rotation y such that y(r) and y(s) are parallel to the real and iaginary axis respectively and R(y(z 1 )) > R(y(z 0 )), that is, we take h [0, 2p) such that I(e ih z)=constant for z r, R(e ih z)=constant for z s and R(e ih z 1 )> R(e ih z 0 ). Then we have that for z Co(D) and R(e ih (z z 0 )) \ R(e ih (z 1 z 0 ))>0 d(z 0, Co(D))=inf{R(e ih (z z 0 )): z Co(D)}. We now take a=e ih in (2.2) and proceed as before. L We now give an exaple proving that (1.3) is sharp. Indeed, let us consider the Sobolev inner product Op, qp=f a p(t) q(t) dt+mp(1) q(1)+npœ(1) qœ(1), 0 where 0<a<1, M,N>0 (see [AMRR1] for an study of this type of inner product). With the notation of Theore 1.1 we have N=1, = q [0, a] dt+d 1, D=[0, a] 2 {1}, Co(D)=[0, 1], w 0 (t)=q [0, a] (t)+mq {1} (t), w 1 (t)=nq {1} (t). Hence, Theore 1.1 gives the bound d(z 0, Co(D)) [ (1/2) `N/M. We now copute an orthogonal polynoial of degree two with respect to O,P. To do that we need the following data: O1, 1P=a+M, Ot, 1P=a 2 /2+M, Ot 2,1P=a 3 /3+M Ot, tp=a 3 /3+M+N, Ot 2,tP=a 4 /4+M+2N. We then have that a+m p 2 (t)=: : a2 /2+M a3 /3+M a 2 /2+M a 3 /3+M+N a 4 /4+M+2N 1 t t 2 is the orthogonal polynoial of degree two with respect to O,P (up to a nonzero ultiplicative constant). Hence, the polynoial 1+M/a a/2+m/a q 2 (t)=: : a2 /3+M/a a/2+m/a a 2 /3+M/a+N/a a 3 /4+M/a+2N/a 1 t t 2 is also orthogonal with respect to O,P.

9 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 135 We take M=N=ea, e >0. The bound given by Theore 1.1 is now (3.6) d(z 0, Co(D)) [ 1/2. Taking the liit as a tends to 0, we get that the largest zero of q 2 tends to the largest zero of the polynoial :1+e e e e 2e 3e 1 t t 2 :, or what is equivalent, to the largest zero of the polynoial :1+e e e t t 2 :. Taking again the liit as e tends to 0, we get that the largest zero of q 2 tends to the largest zero of the polynoial : t t 2 :=t(2t 3), that is, the largest zero of q 2 tends to 3/2, and since Co(D)=[0, 1], we deduce that (3.6) is sharp. We can proceed in a slightly different way than in the proof of Theore 1.1 to obtain another bound on the zeros: Theore 3.2. Let 0<a i, b i <2 satisfy a i +b i =2, i=1,..., N 1, and a 0 =2, b N =2. Then, with the hypotheses of Theore 1.1 we also have (3.7) d(z 0, Co(D)) [ ax3 k `C k `a k 1 b k, k=1,..., N4. Proof. We again assue (3.2) and that Rz Rz 0 > Rz 1 Rz 0 >0 for z Co(D). To prove the estiate (3.7) we split up the atrix C(z, z 0 ) W(z)+ W(z) C(z, z 0 ) g (see (3.3)) as follows: C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g =X 1 +X 2 + +X N,

10 136 DURAN AND SAFF where z0)w0 w w 1 b 1 R(z z 0 )w X 1 =R2R(z S, x x x z x a 1 R(z z 0 )w 1 2w w 2 b 2 R(z z 0 )w X 2 =R0 S, x x x z z x x x x z z x X N =R a N 1 R(z z 0 )w N 1 Nw N Nw N 2R(z z 0 )w N Since for soe z D the heritian atrix C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g can not be positive definite, at least one of the atrices X, =1,..., N, should not be positive seidefinite for soe z D. Now (3.7) can be easily deduced taking into account that Rz Rz 0 >0in D. L Our ethod can also be applied to locate the zeros of orthogonal polynoials with respect to general Sobolev inner product of size 2 2. S. Theore 3.3. Consider a Sobolev inner product of the for 1 Op, qp= C F p (k) (z) q (k) (z) w k (z) d(z) k=0 +F (pœ(t) q(t)+p(t) qœ(t)) v(z) d(z),

11 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 137 where is a finite positive Borel easure with support D C (D containing infinitely any points), w 0,w 1,v L 1 (), w 0,w 1 \ 0 and w 0 w 1 v 2 \ 0. Also assue that w 2 1/(w 0 w 1 v 2 ) L. (). If z 0 is a zero of an orthogonal polynoial with respect to O,P, then (3.8) Proof. d(z 0, Co(D)) [ 1 2 > w 2 1 w 0 w 1 v 2 > 1/2 Indeed, proceeding as before, we find that.. C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g =R 2R(z z 0)w 0 +2v 2R(z z 0 ) v+w 1 2R(z z 0 ) v+w 1 2R(z z 0 )w 1 S (3.8) can be deduced taking into account that that atrix can not be positive definite for soe z D. L. 4. POLYNOMIALS SATISFYING HIGHER-ORDER RECURRENCE RELATIONS In this section we prove Theore 1.2. Indeed, taking into account the definition of the operators T (see (1.5)) we get for =1,..., N, that and for =0 that Fro this we find: T ((z w) p(z))= wt (p(z))+t 1 (p(z)), T 0 ((z w) p(z))= wt 0 (p(z))+zt N (p(z)) w w C(z, w)=r w x x x z z x w 1 z w S.

12 138 DURAN AND SAFF Proceeding as in the previous section we obtain: C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g w z w0 w 1 2R(z 0 )w 1 w =R 2R(z0)w0 0 w 2 2R(z 0 )w S. x x z z x x zw w N 2R(z 0 )w N We now split up the heritian atrix C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g follows: as C(z, z 0 ) W(z)+W(z) C(z, z 0 ) g =X+X 1 +X 2 + +X N, where R(z0)w z w X=R x x z x x S, zw R(z 0 )w N R(z0)w0 w w 1 R(z 0 )w X 1 =R x x x z x R(z 0 )w 1 w w 2 R(z 0 )w X 2 =R0 S, x x x x z x x S,

13 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS x z x x x X N =R S R(z 0 )w N 1 w N w N (Rz 0 )w N And so, if Rz 0 <0 at least one of the atrices X, =1,..., N, orx cannot be positive seidefinite (respectively negative seidefinite if Rz 0 >0), fro which (1.7) can be deduced. L We now prove that (1.7) is sharp. To do that we consider a positive easure n whose support is the interval [ 1,1] and write (r n ) n for its sequence of orthonoral polynoials. Hence the set of zeros of (r n ) n is dense in [ 1,1]. We define the sequence of polynoials (p n ) n as follows: p 2n (t)=r n (t 2 ), p 2n+1 (t)=tr n (t 2 ). It is clear that (p n ) n are orthonoral with respect to the inner product (1.6) when N=1, =n, D=[ 1, 1] and w 0 =w 1 =1. Ifz 0 is a zero of soe p n, Theore 1.2 gives that (4.1) z 0 [ 1. Since the set of zeros of (r n ) n is dense in [ 1,1] we can take a sequence of zeros (x k ) k converging to 1 (or 1). Each square root of x k is then a zero of p n for certain n, and then (4.1) is sharp. Finally, we study the general case of size 2 2. Theore 4.1. Consider an inner product of the for (4.2) Op, qp=f [T 0 (p(z)) T 0 (q(z)) w 0 (z)+t 1 (p(z)) T 1 (q(z)) w 1 (z) +(T 0 (p(z)) T 1 (q(z))+t 1 (p(z)) T 0 (q(z))) v(z)] d(z), where the operators T 0 and T 1 are given by (1.5) for N=1, is a finite positive Borel easure with copact support D C, (D containing infinitely any points) and w 0,w 1,v L 1 (), w 0,w 1 \ 0 and w 0 w 1 v 2 \ 0. We assue that, v /w 0, zw 0 (z)+w 1 (z) 2 4R(z) v 2 (z) w 0 (z) w 1 (z) v 2 (z) zw, 0 (z) w 1 (z) 2 w 0 (z) w 1 (z) v 2 (z) L. ().

14 140 DURAN AND SAFF If z 0 is a zero of an orthogonal polynoial with respect to O,P, then (4.3) z 0 [ `C 2 1+C 2 2, where and C 1 =ax3 v /w0., (1/2)> zw 0(z)+w 1 (z) 2 4R(z) v 2 (z) w 0 (z) w 1 > 1/2 (z) v 2 (z) C 2 =(1/2)> zw 0(z) w 1 (z) 2 >1/2. w 0 (z) w 1 (z) v 2 (z) Proof. We assue Rz 0 >0. Taking a= 1 in (2.2) (a=1 if Rz 0 [ 0), and applying our ethod we deduce that for soe z D the heritian atrix R 2Rz 0 w 0 2v 2Rz 0 v (z w 0 +w 1 ) S 2Rz 0 v (zw 0 +w 1 ) 2Rz 0 w 1 2Rzv can not be positive definite. An easy coputation gives that Rz 0 [ v /w 0,or Rz 0 [ 1 2 [ zw 0(z)+w 1 (z) 2 4R(z) v 2 (z)]/[w 0 (z) w 1 (z) v 2 (z)] 1/2, for soe z D. Taking now a=i in (2.2) we deduce that for soe z D the heritian atrix R 2Iz 0 w 0 2Iz 0 v iz w 0 +iw 1 2Iz 0 v+izw 0 iw 1 2Iz 0 w 1 S 2Izv can not be positive definite, fro where we find that Iz 0 [ 1 2 [ zw 0(z) w 1 (z) 2 ]/[w 0 (z) w 1 (z) v 2 (z)] 1/2,.. 4 for soe z D. L REFERENCES [AMRR1] M. Alfaro, F. Marcellan, M. L. Rezola, and A. Ronveaux, On orthogonal polynoials of Sobolev type: algebraic properties and zeros, SIAM J. Math. Anal. 23 (1992),

15 ZEROS OF NONSTANDARD ORTHOGONAL POLYNOMIALS 141 [AMRR2] M. Alfaro, F. Marcellan, M. L. Rezola, and A. Ronveaux, Sobolev type orthogonal polynoials: the non-diagonal case, J. Approx. Theory 83 (1995), [D1] A. J. Duran, A generalization of Favard s Theore for polynoials satisfying a recurrence relation, J. Approx. Theory 74 (1993), [D2] A. J. Duran, On orthogonal polynoials with respect to a positive definite atrix of easures, Canad. J. Math. 47 (1995), [DV] A. J. Duran and W. Van Assche, Orthogonal atrix polynoials and higher order recurrence relations, Linear Algebra Appl. 219 (1995), [GK] W. Gautschi and A. B. J. Kuijlaars, Zeros and critical points of Sobolev orthogonal polynoials, J. Approx. Theory 91 (1997), [LP] G. Lopez Lagoasino and H. Pijeira Cabrera, Zero location and n-th root asyptotics of Sobolev orthogonal polynoials, J. Approx. Theory 99 (1999), [LPP] G. Lopez Lagoasino, I. Perez Izquierdo, and H. Pijeira Cabrera, Sobolev orthogonal polynoials in the coplex plane, J. Cop. Appl. Math., in press. [ST] H. Stahl and V. Totik, General Orthogonal Polynoials, Cabridge Univ. Press, Cabridge, UK, 1992.