On asymptotic properties of FreudâSobolev ... - Semantic Scholar
ARTICLE IN PRESS
Journal of Approximation Theory 125 (2003) 26–41
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On asymptotic properties of Freud–Sobolev orthogonal polynomials Alicia Cachafeiro,a,1 Francisco Marcella´n,b,2 and Juan J. Moreno-Balca´zarc,d,,3 a Departamento de Matema´tica Aplicada, Universidad de Vigo, Spain Departamento de Matema´ticas, Universidad Carlos III de Madrid, Spain c Departamento de Estadı´stica y Matema´tica Aplicada, Universidad de Almerı´a, La Can˜ada de San Urbano s/n, 04120 Almeria, Spain d Instituto Carlos I de Fı´sica Teo´rica y Computacional, Universidad de Granada, Spain b
Received 17 July 2002; accepted in revised form 16 September 2003 Communicated by Walter Van Assche
Abstract In this paper we consider a Sobolev inner product Z Z ð f ; gÞS ¼ fg dm þ l f 0 g0 dm
ðÞ
and we characterize the measures m for which there exists an algebraic relation between the polynomials, fPn g; orthogonal with respect to the measure m and the polynomials, fQn g; orthogonal with respect to ðÞ; such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a 4 Freud weight. In particular, we study the case dm ¼ e x dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud
Corresponding author. Departamento de Estadı´ stica y Matema´tica Aplicada, Universidad de Almerı´ a, La Can˜ada de San Urbano s/n, 04120 Almeria, Spain. Fax: +34-950-015167. E-mail address: [email protected] (J.J. Moreno-Balca´zar). 1 Research partially supported by Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM 2000-0015. 2 Research partially supported by Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM2003-06335-C03-02, by INTAS Project 2000-272 and by the NATO collaborative Grant PST.CLG. 979738. 3 Research partially supported by Junta de Andalucı´ a, Grupo de Investigacio´n FQM 0229, Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM 2001-3878-C0202 and INTAS Project 2000-272.
0021-9045/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2003.09.003
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4
weight e x ) and the Sobolev orthogonal polynomials Qn : Finally, we obtain some asymptotics for fQn g: More precisely, we give the relative asymptotics fQn ðxÞ=Pn ðxÞg on compact subsets pffiffiffi pffiffiffi of C\R as well as the outer Plancherel–Rotach-type asymptotics fQn ð 4 nxÞ=Pn ð 4 nxÞg on ffiffiffiffiffiffiffi ffi p compact subsets of C\½ a; a being a ¼ 4 4=3: r 2003 Elsevier Inc. All rights reserved. MSC: 33C45; 33C47; 42C05 Keywords: Sobolev orthogonal polynomials; Freud polynomials; Asymptotics
1. Introduction The study of algebraic and analytic properties of polynomials orthogonal with respect to an inner product N Z X ð p; qÞS ¼ lk pðkÞ ðxÞqðkÞ ðxÞ dmk ðxÞ; ð1Þ k¼0
R
where ðmk ÞN k¼0 are measures supported on subsets of the real line and p; q are polynomials with real coefficients has attracted the interest of many researchers in the last years (see [10]). Despite the interest of this case (1), the approach was started for N ¼ 1: In such a situation several examples were very carefully analyzed from an algebraic point of view. The first one (see [7]) is related to Gegenbauer measures, i.e., dm0 ðxÞ ¼ dm1 ðxÞ ¼ ð1 x2 Þa w½ 1;1 dx; a 4 1 which represents a situation of measures with compact support. A second one, for unbounded support, is analyzed in [8] when dm0 ðxÞ ¼ dm1 ðxÞ ¼ xa e x wRþ dx; a4 1: In both cases, the basic differences with the so-called standard case ðN ¼ 0Þ are emphasized. In particular, the three-term recurrence relation for the orthogonal polynomials fails and, as a consequence, the study of algebraic and analytic properties of the corresponding sequences of orthogonal polynomials needs different tools. In a more general framework, if m0 or m1 are classical measures (Jacobi, Laguerre, Hermite), then the basic idea is to consider a companion measure which satisfies the so-called coherence condition or symmetric coherence for symmetric measures (see [3]). The goal of coherence is the fact that we can establish a finite algebraic relation between the orthogonal polynomials, fPn g; associated with m0 and the orthogonal polynomials, fQn g; with respect to the inner product (1) for N ¼ 1; the so-called Sobolev orthogonal polynomials. This algebraic relation plays an important role in the study of fQn g since it allows to express the polynomials fQn g in terms of the standard polynomials fPn g and thus it is possible to carry out a study of fQn g from the algebraic, analytic and computational points of view. Notice that in [13] it was proved that if ðm0 ; m1 Þ is a coherent pair of measures, then one of them must be classical and its companion is a perturbation of it. If both measures m0 and m1 have unbounded support then, except for coherent pairs, very few examples are known when m0 and m1 are non-classical measures with
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non-zero absolutely continuous part. However, in the bounded case, i.e., the measures have compact support, quite a few things are known when both measures are non-classical. For instance, a nice survey about asymptotics of Sobolev orthogonal polynomials is [10]. Indeed, one of the aims of our contribution is to analyze orthogonal polynomials for an inner product (1) when dm0 ðxÞ ¼ dm1 ðxÞ ¼ 4 e x wR dx; an example of a non-classical measure. The sequence of standard polynomials orthogonal with respect to such kind of measures has been introduced by Nevai [14,15] in the framework of the so-called Freud measures. They belong to the set of semiclassical measures, i.e., the linear functional u : P-R given by ðu; pÞ ¼ R R pðxÞ dmðxÞ; where P is the linear space of polynomials with real coefficients is such that there exist polynomials f; c with deg cX1 and (see [9]) DðfuÞ ¼ cu:
ð2Þ
Indeed, Freud measures are defined by weight functions wðxÞ ¼ e P where P is a monic polynomial of degree 2n: For Sobolev inner products (1) with N ¼ 1 and m0 ¼ m1 ¼ m the following result is proved in [2]. Proposition 1. If m is a semiclassical measure such that (2) holds, then there exists a non-negative integer number s such that nþs0 X
an; n s a0;
ð3Þ
In what follows, we use the inner product Z Z ð p; qÞS ¼ pq dm þ l p0 q0 dm; l40;
ð4Þ
fðxÞPn ðxÞ ¼
an; j Qj ðxÞ;
j¼n s
where deg f ¼ s0 :
R
R
and we denote by fPn g the sequence of orthogonal polynomials associated with m ¼ m0 ¼ m1 and by fQn g the sequence of orthogonal polynomials with respect to (4). We are interested in a converse result of Proposition 1, i.e., if we consider an inner product (4), such that (3) holds, then the goal is to know what information about the measure m can be given. Indeed, we get Theorem 1. Relation (3) holds if and only if the measure m is semiclassical. Furthermore, the polynomials f; c in (2) can explicitly be given. In particular, if f 1 then m is a Freud measure. Our second step is to analyze orthogonal polynomials associated with the Sobolev 4 inner product (4) when dm ¼ e x dx: In Section 3 we deduce the connection between the sequences fPn g and fQn g: In such a way we can obtain an explicit expression for fQn g in terms of fPn g: From it we deduce in Section 4 the relative asymptotics of Qn
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with respect to Pn as well as a Plancherel–Rotach-type asymptotics formula for Qn : Here the scaling in the variable is needed.
2. Proof of Theorem 1 Let f be a polynomial of degree s0 such that fðxÞPn ðxÞ ¼
nþs0 X
an; j Qj ðxÞ;
ð5Þ
nXs;
j¼n s
with an;n s a0 and s0 ps: From (5), we get 0 ¼ ðfðxÞPn ðxÞ; Qj ðxÞÞS ;
j ¼ 0; 1; y; n s 1;
ð6Þ
0aðfðxÞPn ðxÞ; Qn s ðxÞÞS ¼ an;n s ðQn s ; Qn s ÞS : From (6),
Z
ð7Þ
Z
ðfðxÞPn ðxÞÞ0 Q0j ðxÞ dm Z Z 0 0 Pn ðxÞ½fðxÞQj ðxÞ þ lf ðxÞQj ðxÞ dm þ l fðxÞP0n ðxÞQ0j ðxÞ dm: ¼
0¼
fðxÞPn ðxÞQj ðxÞ dm þ l
R
R
R
R
The degree of the polynomial inside the brackets is s0 þ j and taking into account that 0pjpn s 1; from the orthogonality of Pn with respect to m we deduce Z fðxÞP0n ðxÞQ0j ðxÞ dm ¼ 0; for 0pjpn s 1: R
This means that the polynomial fP0n is orthogonal to Pn s 2 with respect to the measure m; i.e., fðxÞP0n ðxÞ
¼
n 1þs X0
bn; j Pj ðxÞ:
ð8Þ
j¼n s 1
On the other hand, from (7), Z Z 0a Pn ðxÞ½fðxÞQn s þ l f0 ðxÞQ0n s ðxÞ dm þ l fðxÞP0n ðxÞQ0n s ðxÞ dm: R
R
The degree of the polynomial inside the brackets is n s þ s0 pn: If s0 os; then we get Z fðxÞP0n ðxÞQ0n s ðxÞ dma0: R
Taking into account (8) and the fact that Q0n s ðxÞ ¼ ðn sÞPn s 1 ðxÞ þ lower degree terms;
ð9Þ
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we get Z R
fðxÞP0n ðxÞPn s 1 ðxÞ dma0;
i.e., in (8) bn;n s 1 a0: Now, if s0 ¼ s; Z Z 0aa P2n ðxÞ dm þ l fðxÞP0n ðxÞQ0n s ðxÞ dm; R
R
where a is the leading coefficient of fðxÞ: Again, from (8) and (9), we get Z Z 2 0aa Pn ðxÞ dm þ lðn sÞbn;n s 1 P2n s 1 ðxÞ dm: R
R
In other words, (7) becomes an;n s ðQn s ; Qn s ÞS ¼ a
Z R
P2n ðxÞ dm
þ lðn sÞbn;n s 1
Z R
P2n s 1 ðxÞ dm:
Thus, bn;n s 1 a0 if and only if Z an;n s ðQn s ; Qn s ÞS aa P2n ðxÞ dm: R
ðu; fðxÞP0n ðxÞÞ
Finally, we have ¼ 0; for nXs þ 2; and then we get ðfðxÞu; P0n ðxÞÞ ¼ 0; for nXs þ 2; i.e., ðDðfuÞ; Pn ðxÞÞ ¼ 0; for nXs þ 2: Thus Dðf uÞ ¼
sþ1 X
bk
k¼0
Pk ðxÞu ; ðu; P2k ðxÞÞ
where bk ¼ ðDðfuÞ; Pk ðxÞÞ ¼ ðfu; P0k ðxÞÞ ¼ ðu; fP0k ðxÞÞ ! s0 X þk 1 bk; j Pj ðxÞ ¼ bk;0 : ¼ u; j¼0
Then, DðfuÞ ¼
sþ1 X bk;0 Pk ðxÞ u ¼ cu; ðu; P2k ðxÞÞ k¼0
where cðxÞ ¼
sþ1 sþ1 X X ðu; fðtÞP0k ðtÞPk ðxÞÞ bk;0 Pk ðxÞ ¼ 2 ðu; Pk ðxÞÞ ðu; P2k ðtÞÞ k¼0 k¼0 ð1;0Þ
¼ ðu; fðtÞKsþ1 ðt; xÞÞ: Here, Kn ðt; xÞ ¼
n X Pk ðtÞPk ðxÞ ðu; P2k ðxÞÞ j¼0
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is the nth kernel polynomial associated with the sequence ðPn Þ and we denote ð1;0Þ @ Kn ðt; xÞ ¼ @t Kn ðt; xÞ: Notice that deg ðfÞ ¼ s0 and 1pdeg ðcÞps þ 1: According to the definition of the order of a semiclassical linear functional (see [9]), the order of u is, at most, maxfs0 2; sg: & The simplest case corresponds to fðxÞ ¼ 1: In this situation ð1;0Þ
Du ¼ ðu; Ksþ1 ðt; xÞÞu;
with sX0:
If u is induced by an absolutely continuous measure m; i.e., dmðxÞ ¼ wðxÞ dx; then R 0 w ðxÞ ¼ cðxÞwðxÞ and wðxÞ ¼ exp cðxÞ dx : Thus, we obtain a Freud weight.
4
3. The Freud weight e x and the Sobolev orthogonal polynomials Let fPn g be the sequence of monic polynomials orthogonal with respect to the 4 weight function dm ¼ e x dx supported on R: As we mentioned in Section 1, they have been considered by Nevai [14,15]. These polynomials satisfy a three-term recurrence relation xPn ðxÞ ¼ Pnþ1 ðxÞ þ cn Pn 1 ðxÞ;
nX1;
with initial conditions P0 ðxÞ ¼ 1 and P1 ðxÞ ¼ x; where the parameters cn satisfy a non-linear recurrence relation (see [4]) n ¼ 4cn ðcnþ1 þ cn þ cn 1 Þ;
ð10Þ
nX1;
with c0 ¼ 0 and c1 ¼ Gð3=4Þ=Gð1=4Þ: R On the other hand, from (8) with s0 ¼ 0 ðf 1Þ and cðxÞ dx ¼ x4 ; i.e., cðxÞ ¼ 4x3 ðs ¼ 2Þ; the polynomials fPn g satisfy a structure relation P0n ðxÞ ¼ nPn 1 ðxÞ þ dn Pn 3 ðxÞ;
nX3;
where RN RN 0 4 4 P ðxÞPn 3 ðxÞe x dx N Pn ðxÞ½P0n 3 ðxÞ 4x3 Pn 3 ðxÞ e x dx N R Nn 2 R ¼ N 2 x4 dx x4 dx N Pn 3 ðxÞe N Pn 3 ðxÞe RN 2 4 4 P ðxÞe x dx ¼ 4cn cn 1 cn 2 ; nX3: ¼ R N N 2 n x4 dx N Pn 3 ðxÞe
dn ¼
We consider the Sobolev inner product Z N Z 4 pðxÞqðxÞe x dx þ l ð p; qÞS ¼ N
N
4
p0 ðxÞq0 ðxÞe x dx;
p; qAP;
N
and let fQn g be the corresponding sequence of monic orthogonal polynomials. Taking into account Proposition 1 as well as the fact that Qn ð xÞ ¼ ð 1Þn Qn ðxÞ we get
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Proposition 2. The polynomial fPn g and fQn g are related by Pn ðxÞ ¼ Qn ðxÞ þ ln 2 Qn 2 ðxÞ;
ð11Þ
nX3:
Proof. From Pn ðxÞ ¼ Qn ðxÞ þ
n 2 X
ln; j Qj ðxÞ;
j¼0
for 0pjpn 2; we get ln; j ¼ ¼
ðPn ðxÞ; Qj ðxÞÞS jjQj jj2S
l
RN
¼
l
RN
4
P0n ðxÞQ0j ðxÞe x dx
N
jjQj jj2S 4
N
4Pn ðxÞx3 Q0j ðxÞe x dx jjQj jj2S
:
This expression vanishes for jon 2: For j ¼ n 2 we get RN 2 4 P ðxÞe x dx ln;n 2 :¼ ln 2 ¼ 4lðn 2Þ N n 40: & jjQn 2 jj2S
ð12Þ
On the other hand, we can observe that Qi ðxÞ ¼ Pi ðxÞ; i ¼ 0; 1; 2: Notice that jjPn jj2S ¼ jjQn jj2S þ l2n 2 jjQn 2 jj2S ; with jjPn jj2S ¼
Z
Z N Z N 4 4 4 P2n ðxÞe x dx þ l ðP0n ðxÞÞ2 e x dx ¼ P2n ðxÞe x dx N N N
Z N Z N 4 2 2 x 2 2 x4 Pn 1 ðxÞe dx þ dn Pn 3 ðxÞe dx ; þl n N
N
ð13Þ
N
and using (12) we have jjQn jj2S þ l2n 2 jjQn 2 jj2S ! RN Z N 4 n N P2nþ2 ðxÞe x dx 2 x4 ¼ 4l þ ðn 2Þ ln 2 Pn ðxÞe dx : ln N Gathering (13) and (14) we obtain Z N Z N Z 4 2 x4 Pn ðxÞe dx þ l n2 P2n 1 ðxÞe x dx þ dn2 N
N
¼ 4l
n ln
Z
N
N
4
P2nþ2 ðxÞe x dx þ ðn 2Þln 2
Z
N
N
N N
4 P2n 3 ðxÞe x
4 P2n ðxÞe x dx :
ð14Þ
dx
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RN 4 Then, dividing by N P2n ðxÞe x dx we get 2
n dn2 n 1þl þ cnþ2 cnþ1 þ ðn 2Þln 2 ; ¼ 4l ln cn cn cn 1 cn 2 or, equivalently, 2
n n þ 16cn cn 1 cn 2 ¼ 4l cnþ2 cnþ1 þ ðn 2Þln 2 ; 1þl ln cn
nX3:
Finally,
1 n2 n þ þ 16cn cn 1 cn 2 ¼ 4 cnþ2 cnþ1 þ ðn 2Þln 2 ; l cn ln
nX3;
ð15Þ
with initial conditions 4c3 c2 c1 ; 1 þ c1 l 1 8c4 c3 c2 : l2 ¼ 4 þ c2 l 1 l1 ¼
Notice that for n ¼ 2m; an even non-negative integer number, we can assume l2m ¼
qm 1 ðl 1 Þ ; qm ðl 1 Þ
mX1;
where qm is a polynomial of degree m: Thus, for mX1; (15) becomes c2mþ2 c2mþ1
8mqm ðl 1 Þ qm 2 ðl 1 Þ 4m2 1 þ 4ð2m 2Þ ¼ l þ þ 16c2m c2m 1 c2m 2 ; c2m qm 1 ðl 1 Þ qm 1 ðl 1 Þ
or, equivalently, qm ðl 1 Þ ¼
qm 1 ðl 1 Þ 4m2 l 1 þ þ 16c2m c2m 1 c2m 2 8mc2mþ2 c2mþ1 c2m
1 m 1 qm 2 ðl Þ : m c2mþ2 c2mþ1
If q˜ m denotes the monic polynomial associated with qm ; i.e., qm ¼ sm q˜ m ; we get
4m2 1 1 þ 16c2m c2m 1 c2m 2 q˜ m 1 ðl 1 Þ q˜ m ðl Þ ¼ l þ c2m 64ðm 1Þ2 c2m c2m 1 q˜ m 2 ðl 1 Þ; with initial conditions q˜ 0 ðl 1 Þ ¼ 1
and
q˜ 1 ðl 1 Þ ¼ l 1 þ
4 : c2
mX2;
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On the other hand, for n ¼ 2m 1; an odd nonnegative integer number, we can assume l2m 1 ¼
rm 1 ðl 1 Þ ; rm ðl 1 Þ
mX1;
where rm is a polynomial of degree m: Thus, for mX2; (15) becomes c2mþ1 c2m
4ð2m 1Þrm ðl 1 Þ rm 2 ðl 1 Þ ð2m 1Þ2 1 þ 4ð2m 3Þ ¼ l þ c2m 1 rm 1 ðl 1 Þ rm 1 ðl 1 Þ þ 16c2m 1 c2m 2 c2m 3 ;
or, equivalently, rm 1 ðl 1 Þ ð2m 1Þ2 rm ðl Þ ¼ l 1 þ þ 16c2m 1 c2m 2 c2m 3 4ð2m 1Þc2mþ1 c2m c2m 1
!
1
2m 3 rm 2 ðl 1 Þ: ð2m 1Þc2mþ1 c2m
If rm ¼ tm r˜m where r˜m denotes the monic polynomial associated with rm ; then we get ! ð2m 1Þ2 1 1 r˜m ðl Þ ¼ l þ þ 16c2m 1 c2m 2 c2m 3 r˜m 1 ðl 1 Þ c2m 1 16ð2m 3Þ2 c2m 1 c2m 2 r˜m 2 ðl 1 Þ;
mX2;
with initial conditions r˜0 ðl 1 Þ ¼ 1
and
r˜1 ðl 1 Þ ¼ l 1 þ
1 : c1
As a conclusion, fq˜ m g and f˜rm g are sequences of monic orthogonal polynomials.
4. Asymptotics of Qn First, we establish the asymptotic behavior of the sequence fln g which appears in (11). pffiffiffi Proposition 3. For the sequence fln = ng we get the upper bound pffiffiffi 5 ln pffiffiffio ; nX1: n 3 pffiffiffi Furthermore, the sequence fln = ng is convergent and ln 1 lim pffiffiffi ¼ pffiffiffi: n 6 3
n-N
ð16Þ
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Proof. By the extremal property of jjQn jj2S we have Z N Z N 4 2 2 x4 2 Pn ðxÞe dx þ ln P2n 1 ðxÞe x dx: jjQn jjS X N
N
Thus, for nX1; from (11) RN 2 4 pffiffiffi N Pnþ2 ðxÞe x dx ln pffiffiffi ¼ 4l n n jjQn jj2S RN 2 4 pffiffiffi Pnþ2 ðxÞe x dx N R p 4l n R N 2 x4 dx þ ln2 N P2 ðxÞe x4 dx N Pn ðxÞe N n 1 pffiffiffi cnþ2 cnþ1 cn ¼ 4l n : cn þ ln2
ð17Þ
From the recurrence relation (10) we have, for nX2; pffiffiffi n 2 4cn ¼ n 4cn ðcnþ1 þ cn 1 Þon ) cn o ; 2 but simple computations show that the above inequality also holds for n ¼ 1; that is, pffiffiffi n ð18Þ cn o ; nX1: 2 Now, using this inequality in (10) we obtain n 3 pffiffiffiffiffiffiffiffiffiffiffi ¼ cn ðcnþ1 þ cn þ cn 1 Þo n þ 1c n 4 2 pffiffiffiffiffiffi nþ1 and, then cn 4n6ðnþ1Þ ; for nX2; but again simple computations prove that this inequality is true for n ¼ 1; and, therefore, n cn 4 pffiffiffiffiffiffiffiffiffiffiffi; nX1: ð19Þ 6 nþ1 We use relations (18) and (19) in (17) obtaining, for nX2; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðn þ 2Þðn þ 1Þ 5 ln l ðn þ 2Þðn þ 1Þ pffiffiffio p ; nX3; o 1 pffiffiffiffiffiffi þ 2ln 3 2n n 3 nþ1 and straightforward computations in (12) show that this inequality holds for n ¼ 1; 2: Thus, pffiffiffi ln 5 pffiffiffio o1; nX1: ð20Þ n 3 On the other hand, relation (15) can be rewritten as ln ¼ 1
4ncnþ2 cnþ1 2
n l þ cn þ 16cn cn 1 cn 2 4ðn 2Þln 2
;
nX3
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and from here we get for nX3; ln pffiffiffi ¼ n
1 pffiffi 1 4l ncnþ2 cnþ1
þ
1 n3=2 4ðcn cnþ1 cnþ2
þ
16c pffiffin cn 1 cn 2 Þ ncnþ1 cnþ2
n 2 cnþ1 cnþ2
qffiffiffiffiffiffi
:
n 2 plffiffiffiffiffiffi n 2 n n 2
ð21Þ
Denoting
1 1 n3=2 16cn cn 1 cn 2 þ þ pffiffiffi BðnÞ ¼ pffiffiffi ; 4l ncnþ2 cnþ1 4 cn cnþ1 cnþ2 ncnþ1 cnþ2 n 2 CðnÞ ¼ cnþ1 cnþ2
rffiffiffiffiffiffiffiffiffiffiffi n 2 ; n
ð22Þ
ð23Þ
then (21) becomes ln 1 pffiffiffi ¼ : n 2 n BðnÞ CðnÞplffiffiffiffiffiffi
ð24Þ
n 2
Taking into account that in [4] an asymptotic expansion of cn was established, in particular cn 1 lim pffiffiffi ¼ pffiffiffi; n 2 3
ð25Þ
n-N
using (25) we get pffiffiffi 20 3 ; lim BðnÞ ¼ n-N 3
lim CðnÞ ¼ 12:
n-N
ð26Þ
pffiffiffi Therefore, if the sequence fln = ng converges, its limit r must satisfy the equation pffiffiffi pffiffiffi pffiffiffi r ¼ 1=ð20 3=3 12rÞ; i.e., either r ¼ 1=6 3 or r ¼ 3=2: But, from (20), it is p ffiffi ffi pffiffiffi deduced that the limit of fln = ng; if it exists, is 1=6 3: Then, to conclude the proof pffiffiffi of this Proposition it only remains to prove that fln = ng converges. pffiffi pffiffiffi If r ¼ 1=6 3 and y ¼ 35; then, from (24), we have n 2 r2 CðnÞ þ r2 CðnÞj ln j1 rBðnÞ þ rCðnÞplffiffiffiffiffiffi n 2 pffiffiffi r ¼ : n n 2 BðnÞ CðnÞplffiffiffiffiffiffi
ð27Þ
n 2
ln On the other hand, using inequality (20) for the sequence pffiffiffi; we have n n 2 oyCðnÞ and so, for n large enough, CðnÞplffiffiffiffiffiffi n 2 ln 2 BðnÞ CðnÞ pffiffiffiffiffiffiffiffiffiffiffi4BðnÞ yCðnÞ40: n 2
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From here we can give a bound for (27), i.e., for n large enough, n 2 rÞ þ r2 CðnÞj j1 rBðnÞ þ r CðnÞðplffiffiffiffiffiffi ln n 2 pffiffiffi ro n BðnÞ yCðnÞ j1 rBðnÞ þ r2 CðnÞj rCðnÞ ln 2 : p ffiffiffiffiffiffiffiffiffiffiffi þ BðnÞ yCðnÞ p r BðnÞ yCðnÞ n 2 Now, taking into account the limits of the sequences BðnÞ and CðnÞ given in (26), we obtain pffiffiffi ln ln 2 2= 3 pffiffiffi lim suppffiffiffiffiffiffiffiffiffiffiffi r lim suppffiffiffi rp 20 pffiffi 4 5 n-N n n-N n 2 3 ln 2 1 pffiffiffiffiffi lim suppffiffiffiffiffiffiffiffiffiffiffi r; ¼ 10 2 15 n-N n 2 where 1 1 pffiffiffiffiffio ; 10 2 15 2 pffiffiffi and, therefore, we can conclude that the sequence fln = ng is convergent and its limit pffiffiffi is r ¼ 1=ð6 3Þ: & We want to compare the asymptotic behavior of Qn and Pn in the complex plane, more exactly, in C\R: We get the following result: Theorem 2. The asymptotic behavior lim
n-N
Qn ðxÞ 3 ¼ Pn ðxÞ 2
ð28Þ
holds uniformly on compact subsets of C\R: Proof. We consider the orthonormal polynomials pn with respect to the inner RN 4 product N f ðxÞgðxÞe x dx: In [5] Lo´pez and Rakhmanov give the strong asymptotics of pn ; i.e., it holds uniformly on compact subsets of C\R; lim
n-N
pn ðxÞ Dn ðxÞðjðx=xn ÞÞ
nþ1=2
1 ¼ pffiffiffiffiffiffi; 2p
+ function for the weight e x on the segment ½ xn ; xn ; i.e., where Dn ðxÞ is the Szego’s (pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ) x2 x2n xn t4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt ; Dn ðxÞ ¼ exp 2 2 2p xn ðx tÞ xn t pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=4 and jðxÞ ¼ x þ x2 1 is the conformal mapping of C\½ 1; 1 being xn ¼ ð4n 3Þ onto the exterior of the closed unit disk. 4
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Thus, we can deduce that lim
n-N
pn ðxÞ ¼ 1; pnþ2 ðxÞ
uniformly on compact subsets of C\R: Then, for the monic polynomials Pn we get pffiffiffi pffiffiffi nPn ðxÞ ¼ 2 3; ð29Þ lim n-N Pnþ2 ðxÞ uniformly on compact subsets of C\R: Dividing relation (11) by Pn ðxÞ we obtain Qn ðxÞ Pn 2 ðxÞ Qn 2 ðxÞ ¼ 1 ln 2 ; Pn ðxÞ Pn ðxÞ Pn 2 ðxÞ where, using (16) and (29), we have lim ln 2
n-N
Pn 2 ðxÞ 1 ¼ ; Pn ðxÞ 3
uniformly on compact subsets of C\R: Now, standard arguments allow us to conclude that the sequence fQn =Pn g is convergent and its limit is the solution of the equation s ¼ 1 þ s=3; that is, s ¼ 3=2: & From this theorem we deduce that the Sobolev polynomials fQn g have the same asymptotic behavior (up to multiplicative constant factors) as fPn g in C\R: This occurs in other cases when the measures m0 and m1 involved in the Sobolev inner product (1) with N ¼ 1 have unbounded support (see [1,6]) but this is not the case when the measures have compact support (see, for example, [11] or [12]). Three natural questions arise. The first one is why does it occur? The second one is when does it occur? Finally, can we give a more complete description of the asymptotic behavior of the polynomials Qn ? The answer to the first and second questions is yet open for us, but we can obtain better information about the asymptotics of Qn if we scale the variable x in a convenient way, i.e., if we look for the exterior Plancherel– Rotach-type asymptotics for Qn : We have the following result: Theorem 3. The asymptotic behavior qffiffi p ffiffiffi 2 4 3 3j 4 4x Qn ð nxÞ p q ffiffi ffiffi ffi lim ¼ n-N Pn ð 4 nxÞ 3j2 4 3 x þ 1
ð30Þ
4
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi holds uniformly on compact subsets of C\½ 4 4=3; 4 4=3 ; where jðxÞ ¼ x þ x2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi with x2 140 if x41; i.e., the conformal mapping of C\½ 1; 1 onto the exterior of the closed unit disk. Proof. It is well-known (see [16]) that from the three-term recurrence relation xpn ðxÞ ¼ anþ1 pnþ1 ðxÞ þ an pn 1 ðxÞ;
nX1;
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we can obtain asymptotic properties of the orthonormal polynomials pn : Indeed, as an 1 ffiffiffiffiffi; ffiffiffiffiffiffiffiffiffiffi ¼ p lim p 4 4 nþj 12
n-N
for a jAR fixed;
we deduce (see [16]) pffiffiffiffiffiffiffiffiffiffi pn 1 ð 4 n þ j xÞ 1 p ffiffiffiffiffiffiffiffiffiffi lim ¼ qffiffi ; n-N pn ð 4 n þ j xÞ 4 3 j x
j fixed;
4
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Then, for the monic polynomial Pn we have p ffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4 p ffiffiffi Pn 1 ð 4 n þ j xÞ 12 4 pffiffiffiffiffiffiffiffiffiffi ð31Þ lim n ¼ qffiffi ; j fixed; n-N 4 3 Pn ð 4 n þ j xÞ j x 4
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Introducing the change of ffiffi ffi p variable x- 4 nx in (11) and using this relation in a recursive way, we get ½ðn 1Þ=2 X p pffiffiffi ffiffiffi ðnÞ 4 Qn ð n xÞ ¼ ð 1Þj b2j Pn 2j ð 4 n xÞ;
nX3;
j¼0
Q ðnÞ ðnÞ with b0 ¼ 1; b2j ¼ ji¼1 ln 2i ; and ½a denotes the integer part of a: Then, dividing pffiffiffi by Pn ð 4 n xÞ we obtain pffiffiffi Qn ð 4 n xÞ ffiffiffi ¼ p Pn ð 4 n xÞ ¼
½ðn 1Þ=2 X j¼0 ½ðn 1Þ=2 X j¼0
pffiffiffi Pn 2j ð 4 n xÞ ffiffiffi p Pn ð 4 n xÞ Qj pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðnÞ b2j n 2i Pn 2j ð 4 n xÞ j p ffiffi ffi ð 1Þ Qj pffiffiffiffiffiffiffiffiffiffiffiffiffi i¼1 ; Pn ð 4 n xÞ n 2i i¼1 ðnÞ
ð 1Þj b2j
where an empty product is equal to 1. Now, we analyze the asymptotic behavior of the factors in the above sum. If we use (16) in Proposition 3 and (31), then
ðnÞ j Y b2j ln 2i 1 j pffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð 1Þj 6 3 n 2i n 2i i¼1 i¼1
lim ð 1Þj Qj
n-N
pffiffiffi j pffiffiffiffiffiffiffiffiffiffiffiffiffi Y Pn 2j ð 4 n xÞ pffiffiffi n 2i lim n-N Pn ð 4 n xÞ i¼1 0 1j pffiffiffi B 2 3 C ¼ @ qffiffi A ; j fixed: j2 4 34 x
j fixed;
ð32Þ
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pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi This last limit holds uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Gathering the above limits we get 0 1j p ffiffi ffi 4 1 B C ðnÞ Pn 2j ð n xÞ p ffiffiffi lim ð 1Þj b2j ð33Þ ¼@ qffiffi A ; 4 n-N 4 3 Pn ð n xÞ 2 3j x 4 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : pffiffiffi On the other hand, the upper bound of the sequence fln = ng obtained in Proposition 3 together with the limit relation (32) allow us to give a uniform bound pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ðnÞ for ð 1Þj b2j Pn 2j ð 4 nxÞ=Pn ð 4 nxÞ on C\½ 4 4=3; 4 4=3 ; that is, for n large enough and 0pjp½ðn 1Þ=2 ; pffiffiffi ð 4 n xÞ ð 1Þj bðnÞ Pn 2jp ffiffiffi pKyj ; 2j Pn ð 4 n xÞ where
pffiffiffi 5 o1 y¼ 3
pffiffiffi pffiffiffi and K is a constant. Therefore, we have a majorant for Qn ð 4 n xÞ=Pn ð 4 n xÞ with ffiffiffiffiffiffiffi ffi p p ffiffiffiffiffiffiffi ffi xAC\½ 4 4=3; 4 4=3 : From Lebesgue’s dominated convergence theorem together with (33) we get 0 1j qffiffi p ffiffiffi 2 4 3 N 3j 4 X 4x Qn ð n xÞ 1 B C p q ffiffi q ffiffi ffiffi ffi lim ¼ ¼ ; @ A n-N Pn ð 4 n xÞ 4 3 j¼0 3j2 3j2 4 34 x þ 1 4x pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 and thus the statement of our qffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi theorem follows. Note that 1= 3j2 4 34 x o1 when xAC\½ 4 4=3; 4 4=3 : &
Acknowledgments We thank Professors Jeff Geronimo and Arno Kuijlaars for their suggestions which have improved the presentation of the paper.
References [1] M. Alfaro, J.J. Moreno-Balca´zar, T.E. Pe´rez, M.A. Pin˜ar, M.L. Rezola, Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs, J. Comput. Appl. Math. 133 (2001) 141–150. [2] W.D. Evans, L.L. Littlejohn, F. Marcella´n, C. Markett, A. Ronveaux, On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal. 26 (2) (1995) 446–467.
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[3] A. Iserles, P.E. Koch, S.P. N^rsett, J.M. Sanz–Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (2) (1991) 151–175. [4] J.S. Lew, D.A. Quarles Jr., Nonnegative solutions of a nonlinear recurrence, J. Approx. Theory 38 (1983) 357–379. [5] G. Lo´pez, E.A. Rakhmanov, Rational approximations, orthogonal polynomials and equilibrium distributions, in: M. Alfaro, et al., (Eds.), Orthogonal Polynomials and their Applications, Lectures Notes in Mathematics, Vol. 1329, Springer, Berlin, 1988, pp. 125–157. [6] F. Marcella´n, J.J. Moreno-Balca´zar, Strong and Plancherel–Rotach asymptotics of non-diagonal Laguerre–Sobolev orthogonal polynomials, J. Approx. Theory 110 (2001) 54–73. [7] F. Marcella´n, T.E. Pe´rez, M.A. Pin˜ar, Gegenbauer–Sobolev orthogonal polynomials, in: A. Cuyt (Ed.), Non-linear Numerical Methods and Rational Approximation, Kluwer Academic Publishers Proceedings, Dordrecht, 1994, pp. 71–82. [8] F. Marcella´n, T.E. Pe´rez, M.A. Pin˜ar, Laguerre–Sobolev orthogonal polynomials, J. Comput. Appl. Math. 71 (1996) 245–265. [9] P. Maroni, Une the´orie alge´brique des polynoˆmes orthogonaux. Application aux polynoˆmes orthogonaux semi-classiques, in: C. Brezinski et al. (Eds.), Orthogonal polynomials and their Applications; IMACS Ann. Comput. Appl. Math. 19 (1991) 95–130. [10] A. Martı´ nez–Finkelshtein, Analytic aspects of Sobolev orthogonality revisited, J. Comput. Appl. Math. 127 (1–2) (2001) 255–266. [11] A. Martı´ nez–Finkelshtein, J.J. Moreno–Balca´zar, Asymptotics of Sobolev orthogonal polynomials for a Jacobi weight, Methods Appl. Anal. 4 (4) (1997) 430–437. [12] A. Martı´ nez–Finkelshtein, J.J. Moreno–Balca´zar, T.E. Pe´rez, M.A. Pin˜ar, Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures, J. Approx. Theory 92 (1998) 280–293. [13] H.G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (3) (1997) 321–343. [14] P. Nevai, Orthogonal polynomials associated with expð x4 Þ; Proc. Canad. Math. Soc. 3 (1983) 263–285. [15] P. Nevai, Asymptotics for orthogonal polynomials associated with expð x4 Þ; SIAM J. Math. Anal. 15 (1984) 1177–1187. [16] W. Van Assche, Asymptotics for Orthogonal Polynomials, in: Lecture Notes in Mathematics, Vol. 1265, Springer, Berlin, 1987.
Journal of Approximation Theory 125 (2003) 26–41
http://www.elsevier.com/locate/jat
On asymptotic properties of Freud–Sobolev orthogonal polynomials Alicia Cachafeiro,a,1 Francisco Marcella´n,b,2 and Juan J. Moreno-Balca´zarc,d,,3 a Departamento de Matema´tica Aplicada, Universidad de Vigo, Spain Departamento de Matema´ticas, Universidad Carlos III de Madrid, Spain c Departamento de Estadı´stica y Matema´tica Aplicada, Universidad de Almerı´a, La Can˜ada de San Urbano s/n, 04120 Almeria, Spain d Instituto Carlos I de Fı´sica Teo´rica y Computacional, Universidad de Granada, Spain b
Received 17 July 2002; accepted in revised form 16 September 2003 Communicated by Walter Van Assche
Abstract In this paper we consider a Sobolev inner product Z Z ð f ; gÞS ¼ fg dm þ l f 0 g0 dm
ðÞ
and we characterize the measures m for which there exists an algebraic relation between the polynomials, fPn g; orthogonal with respect to the measure m and the polynomials, fQn g; orthogonal with respect to ðÞ; such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a 4 Freud weight. In particular, we study the case dm ¼ e x dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud
Corresponding author. Departamento de Estadı´ stica y Matema´tica Aplicada, Universidad de Almerı´ a, La Can˜ada de San Urbano s/n, 04120 Almeria, Spain. Fax: +34-950-015167. E-mail address: [email protected] (J.J. Moreno-Balca´zar). 1 Research partially supported by Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM 2000-0015. 2 Research partially supported by Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM2003-06335-C03-02, by INTAS Project 2000-272 and by the NATO collaborative Grant PST.CLG. 979738. 3 Research partially supported by Junta de Andalucı´ a, Grupo de Investigacio´n FQM 0229, Direccio´n General de Investigacio´n (Ministerio de Ciencia y Tecnologı´ a) of Spain under Grant BFM 2001-3878-C0202 and INTAS Project 2000-272.
0021-9045/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2003.09.003
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4
weight e x ) and the Sobolev orthogonal polynomials Qn : Finally, we obtain some asymptotics for fQn g: More precisely, we give the relative asymptotics fQn ðxÞ=Pn ðxÞg on compact subsets pffiffiffi pffiffiffi of C\R as well as the outer Plancherel–Rotach-type asymptotics fQn ð 4 nxÞ=Pn ð 4 nxÞg on ffiffiffiffiffiffiffi ffi p compact subsets of C\½ a; a being a ¼ 4 4=3: r 2003 Elsevier Inc. All rights reserved. MSC: 33C45; 33C47; 42C05 Keywords: Sobolev orthogonal polynomials; Freud polynomials; Asymptotics
1. Introduction The study of algebraic and analytic properties of polynomials orthogonal with respect to an inner product N Z X ð p; qÞS ¼ lk pðkÞ ðxÞqðkÞ ðxÞ dmk ðxÞ; ð1Þ k¼0
R
where ðmk ÞN k¼0 are measures supported on subsets of the real line and p; q are polynomials with real coefficients has attracted the interest of many researchers in the last years (see [10]). Despite the interest of this case (1), the approach was started for N ¼ 1: In such a situation several examples were very carefully analyzed from an algebraic point of view. The first one (see [7]) is related to Gegenbauer measures, i.e., dm0 ðxÞ ¼ dm1 ðxÞ ¼ ð1 x2 Þa w½ 1;1 dx; a 4 1 which represents a situation of measures with compact support. A second one, for unbounded support, is analyzed in [8] when dm0 ðxÞ ¼ dm1 ðxÞ ¼ xa e x wRþ dx; a4 1: In both cases, the basic differences with the so-called standard case ðN ¼ 0Þ are emphasized. In particular, the three-term recurrence relation for the orthogonal polynomials fails and, as a consequence, the study of algebraic and analytic properties of the corresponding sequences of orthogonal polynomials needs different tools. In a more general framework, if m0 or m1 are classical measures (Jacobi, Laguerre, Hermite), then the basic idea is to consider a companion measure which satisfies the so-called coherence condition or symmetric coherence for symmetric measures (see [3]). The goal of coherence is the fact that we can establish a finite algebraic relation between the orthogonal polynomials, fPn g; associated with m0 and the orthogonal polynomials, fQn g; with respect to the inner product (1) for N ¼ 1; the so-called Sobolev orthogonal polynomials. This algebraic relation plays an important role in the study of fQn g since it allows to express the polynomials fQn g in terms of the standard polynomials fPn g and thus it is possible to carry out a study of fQn g from the algebraic, analytic and computational points of view. Notice that in [13] it was proved that if ðm0 ; m1 Þ is a coherent pair of measures, then one of them must be classical and its companion is a perturbation of it. If both measures m0 and m1 have unbounded support then, except for coherent pairs, very few examples are known when m0 and m1 are non-classical measures with
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non-zero absolutely continuous part. However, in the bounded case, i.e., the measures have compact support, quite a few things are known when both measures are non-classical. For instance, a nice survey about asymptotics of Sobolev orthogonal polynomials is [10]. Indeed, one of the aims of our contribution is to analyze orthogonal polynomials for an inner product (1) when dm0 ðxÞ ¼ dm1 ðxÞ ¼ 4 e x wR dx; an example of a non-classical measure. The sequence of standard polynomials orthogonal with respect to such kind of measures has been introduced by Nevai [14,15] in the framework of the so-called Freud measures. They belong to the set of semiclassical measures, i.e., the linear functional u : P-R given by ðu; pÞ ¼ R R pðxÞ dmðxÞ; where P is the linear space of polynomials with real coefficients is such that there exist polynomials f; c with deg cX1 and (see [9]) DðfuÞ ¼ cu:
ð2Þ
Indeed, Freud measures are defined by weight functions wðxÞ ¼ e P where P is a monic polynomial of degree 2n: For Sobolev inner products (1) with N ¼ 1 and m0 ¼ m1 ¼ m the following result is proved in [2]. Proposition 1. If m is a semiclassical measure such that (2) holds, then there exists a non-negative integer number s such that nþs0 X
an; n s a0;
ð3Þ
In what follows, we use the inner product Z Z ð p; qÞS ¼ pq dm þ l p0 q0 dm; l40;
ð4Þ
fðxÞPn ðxÞ ¼
an; j Qj ðxÞ;
j¼n s
where deg f ¼ s0 :
R
R
and we denote by fPn g the sequence of orthogonal polynomials associated with m ¼ m0 ¼ m1 and by fQn g the sequence of orthogonal polynomials with respect to (4). We are interested in a converse result of Proposition 1, i.e., if we consider an inner product (4), such that (3) holds, then the goal is to know what information about the measure m can be given. Indeed, we get Theorem 1. Relation (3) holds if and only if the measure m is semiclassical. Furthermore, the polynomials f; c in (2) can explicitly be given. In particular, if f 1 then m is a Freud measure. Our second step is to analyze orthogonal polynomials associated with the Sobolev 4 inner product (4) when dm ¼ e x dx: In Section 3 we deduce the connection between the sequences fPn g and fQn g: In such a way we can obtain an explicit expression for fQn g in terms of fPn g: From it we deduce in Section 4 the relative asymptotics of Qn
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with respect to Pn as well as a Plancherel–Rotach-type asymptotics formula for Qn : Here the scaling in the variable is needed.
2. Proof of Theorem 1 Let f be a polynomial of degree s0 such that fðxÞPn ðxÞ ¼
nþs0 X
an; j Qj ðxÞ;
ð5Þ
nXs;
j¼n s
with an;n s a0 and s0 ps: From (5), we get 0 ¼ ðfðxÞPn ðxÞ; Qj ðxÞÞS ;
j ¼ 0; 1; y; n s 1;
ð6Þ
0aðfðxÞPn ðxÞ; Qn s ðxÞÞS ¼ an;n s ðQn s ; Qn s ÞS : From (6),
Z
ð7Þ
Z
ðfðxÞPn ðxÞÞ0 Q0j ðxÞ dm Z Z 0 0 Pn ðxÞ½fðxÞQj ðxÞ þ lf ðxÞQj ðxÞ dm þ l fðxÞP0n ðxÞQ0j ðxÞ dm: ¼
0¼
fðxÞPn ðxÞQj ðxÞ dm þ l
R
R
R
R
The degree of the polynomial inside the brackets is s0 þ j and taking into account that 0pjpn s 1; from the orthogonality of Pn with respect to m we deduce Z fðxÞP0n ðxÞQ0j ðxÞ dm ¼ 0; for 0pjpn s 1: R
This means that the polynomial fP0n is orthogonal to Pn s 2 with respect to the measure m; i.e., fðxÞP0n ðxÞ
¼
n 1þs X0
bn; j Pj ðxÞ:
ð8Þ
j¼n s 1
On the other hand, from (7), Z Z 0a Pn ðxÞ½fðxÞQn s þ l f0 ðxÞQ0n s ðxÞ dm þ l fðxÞP0n ðxÞQ0n s ðxÞ dm: R
R
The degree of the polynomial inside the brackets is n s þ s0 pn: If s0 os; then we get Z fðxÞP0n ðxÞQ0n s ðxÞ dma0: R
Taking into account (8) and the fact that Q0n s ðxÞ ¼ ðn sÞPn s 1 ðxÞ þ lower degree terms;
ð9Þ
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we get Z R
fðxÞP0n ðxÞPn s 1 ðxÞ dma0;
i.e., in (8) bn;n s 1 a0: Now, if s0 ¼ s; Z Z 0aa P2n ðxÞ dm þ l fðxÞP0n ðxÞQ0n s ðxÞ dm; R
R
where a is the leading coefficient of fðxÞ: Again, from (8) and (9), we get Z Z 2 0aa Pn ðxÞ dm þ lðn sÞbn;n s 1 P2n s 1 ðxÞ dm: R
R
In other words, (7) becomes an;n s ðQn s ; Qn s ÞS ¼ a
Z R
P2n ðxÞ dm
þ lðn sÞbn;n s 1
Z R
P2n s 1 ðxÞ dm:
Thus, bn;n s 1 a0 if and only if Z an;n s ðQn s ; Qn s ÞS aa P2n ðxÞ dm: R
ðu; fðxÞP0n ðxÞÞ
Finally, we have ¼ 0; for nXs þ 2; and then we get ðfðxÞu; P0n ðxÞÞ ¼ 0; for nXs þ 2; i.e., ðDðfuÞ; Pn ðxÞÞ ¼ 0; for nXs þ 2: Thus Dðf uÞ ¼
sþ1 X
bk
k¼0
Pk ðxÞu ; ðu; P2k ðxÞÞ
where bk ¼ ðDðfuÞ; Pk ðxÞÞ ¼ ðfu; P0k ðxÞÞ ¼ ðu; fP0k ðxÞÞ ! s0 X þk 1 bk; j Pj ðxÞ ¼ bk;0 : ¼ u; j¼0
Then, DðfuÞ ¼
sþ1 X bk;0 Pk ðxÞ u ¼ cu; ðu; P2k ðxÞÞ k¼0
where cðxÞ ¼
sþ1 sþ1 X X ðu; fðtÞP0k ðtÞPk ðxÞÞ bk;0 Pk ðxÞ ¼ 2 ðu; Pk ðxÞÞ ðu; P2k ðtÞÞ k¼0 k¼0 ð1;0Þ
¼ ðu; fðtÞKsþ1 ðt; xÞÞ: Here, Kn ðt; xÞ ¼
n X Pk ðtÞPk ðxÞ ðu; P2k ðxÞÞ j¼0
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is the nth kernel polynomial associated with the sequence ðPn Þ and we denote ð1;0Þ @ Kn ðt; xÞ ¼ @t Kn ðt; xÞ: Notice that deg ðfÞ ¼ s0 and 1pdeg ðcÞps þ 1: According to the definition of the order of a semiclassical linear functional (see [9]), the order of u is, at most, maxfs0 2; sg: & The simplest case corresponds to fðxÞ ¼ 1: In this situation ð1;0Þ
Du ¼ ðu; Ksþ1 ðt; xÞÞu;
with sX0:
If u is induced by an absolutely continuous measure m; i.e., dmðxÞ ¼ wðxÞ dx; then R 0 w ðxÞ ¼ cðxÞwðxÞ and wðxÞ ¼ exp cðxÞ dx : Thus, we obtain a Freud weight.
4
3. The Freud weight e x and the Sobolev orthogonal polynomials Let fPn g be the sequence of monic polynomials orthogonal with respect to the 4 weight function dm ¼ e x dx supported on R: As we mentioned in Section 1, they have been considered by Nevai [14,15]. These polynomials satisfy a three-term recurrence relation xPn ðxÞ ¼ Pnþ1 ðxÞ þ cn Pn 1 ðxÞ;
nX1;
with initial conditions P0 ðxÞ ¼ 1 and P1 ðxÞ ¼ x; where the parameters cn satisfy a non-linear recurrence relation (see [4]) n ¼ 4cn ðcnþ1 þ cn þ cn 1 Þ;
ð10Þ
nX1;
with c0 ¼ 0 and c1 ¼ Gð3=4Þ=Gð1=4Þ: R On the other hand, from (8) with s0 ¼ 0 ðf 1Þ and cðxÞ dx ¼ x4 ; i.e., cðxÞ ¼ 4x3 ðs ¼ 2Þ; the polynomials fPn g satisfy a structure relation P0n ðxÞ ¼ nPn 1 ðxÞ þ dn Pn 3 ðxÞ;
nX3;
where RN RN 0 4 4 P ðxÞPn 3 ðxÞe x dx N Pn ðxÞ½P0n 3 ðxÞ 4x3 Pn 3 ðxÞ e x dx N R Nn 2 R ¼ N 2 x4 dx x4 dx N Pn 3 ðxÞe N Pn 3 ðxÞe RN 2 4 4 P ðxÞe x dx ¼ 4cn cn 1 cn 2 ; nX3: ¼ R N N 2 n x4 dx N Pn 3 ðxÞe
dn ¼
We consider the Sobolev inner product Z N Z 4 pðxÞqðxÞe x dx þ l ð p; qÞS ¼ N
N
4
p0 ðxÞq0 ðxÞe x dx;
p; qAP;
N
and let fQn g be the corresponding sequence of monic orthogonal polynomials. Taking into account Proposition 1 as well as the fact that Qn ð xÞ ¼ ð 1Þn Qn ðxÞ we get
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Proposition 2. The polynomial fPn g and fQn g are related by Pn ðxÞ ¼ Qn ðxÞ þ ln 2 Qn 2 ðxÞ;
ð11Þ
nX3:
Proof. From Pn ðxÞ ¼ Qn ðxÞ þ
n 2 X
ln; j Qj ðxÞ;
j¼0
for 0pjpn 2; we get ln; j ¼ ¼
ðPn ðxÞ; Qj ðxÞÞS jjQj jj2S
l
RN
¼
l
RN
4
P0n ðxÞQ0j ðxÞe x dx
N
jjQj jj2S 4
N
4Pn ðxÞx3 Q0j ðxÞe x dx jjQj jj2S
:
This expression vanishes for jon 2: For j ¼ n 2 we get RN 2 4 P ðxÞe x dx ln;n 2 :¼ ln 2 ¼ 4lðn 2Þ N n 40: & jjQn 2 jj2S
ð12Þ
On the other hand, we can observe that Qi ðxÞ ¼ Pi ðxÞ; i ¼ 0; 1; 2: Notice that jjPn jj2S ¼ jjQn jj2S þ l2n 2 jjQn 2 jj2S ; with jjPn jj2S ¼
Z
Z N Z N 4 4 4 P2n ðxÞe x dx þ l ðP0n ðxÞÞ2 e x dx ¼ P2n ðxÞe x dx N N N
Z N Z N 4 2 2 x 2 2 x4 Pn 1 ðxÞe dx þ dn Pn 3 ðxÞe dx ; þl n N
N
ð13Þ
N
and using (12) we have jjQn jj2S þ l2n 2 jjQn 2 jj2S ! RN Z N 4 n N P2nþ2 ðxÞe x dx 2 x4 ¼ 4l þ ðn 2Þ ln 2 Pn ðxÞe dx : ln N Gathering (13) and (14) we obtain Z N Z N Z 4 2 x4 Pn ðxÞe dx þ l n2 P2n 1 ðxÞe x dx þ dn2 N
N
¼ 4l
n ln
Z
N
N
4
P2nþ2 ðxÞe x dx þ ðn 2Þln 2
Z
N
N
N N
4 P2n 3 ðxÞe x
4 P2n ðxÞe x dx :
ð14Þ
dx
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RN 4 Then, dividing by N P2n ðxÞe x dx we get 2
n dn2 n 1þl þ cnþ2 cnþ1 þ ðn 2Þln 2 ; ¼ 4l ln cn cn cn 1 cn 2 or, equivalently, 2
n n þ 16cn cn 1 cn 2 ¼ 4l cnþ2 cnþ1 þ ðn 2Þln 2 ; 1þl ln cn
nX3:
Finally,
1 n2 n þ þ 16cn cn 1 cn 2 ¼ 4 cnþ2 cnþ1 þ ðn 2Þln 2 ; l cn ln
nX3;
ð15Þ
with initial conditions 4c3 c2 c1 ; 1 þ c1 l 1 8c4 c3 c2 : l2 ¼ 4 þ c2 l 1 l1 ¼
Notice that for n ¼ 2m; an even non-negative integer number, we can assume l2m ¼
qm 1 ðl 1 Þ ; qm ðl 1 Þ
mX1;
where qm is a polynomial of degree m: Thus, for mX1; (15) becomes c2mþ2 c2mþ1
8mqm ðl 1 Þ qm 2 ðl 1 Þ 4m2 1 þ 4ð2m 2Þ ¼ l þ þ 16c2m c2m 1 c2m 2 ; c2m qm 1 ðl 1 Þ qm 1 ðl 1 Þ
or, equivalently, qm ðl 1 Þ ¼
qm 1 ðl 1 Þ 4m2 l 1 þ þ 16c2m c2m 1 c2m 2 8mc2mþ2 c2mþ1 c2m
1 m 1 qm 2 ðl Þ : m c2mþ2 c2mþ1
If q˜ m denotes the monic polynomial associated with qm ; i.e., qm ¼ sm q˜ m ; we get
4m2 1 1 þ 16c2m c2m 1 c2m 2 q˜ m 1 ðl 1 Þ q˜ m ðl Þ ¼ l þ c2m 64ðm 1Þ2 c2m c2m 1 q˜ m 2 ðl 1 Þ; with initial conditions q˜ 0 ðl 1 Þ ¼ 1
and
q˜ 1 ðl 1 Þ ¼ l 1 þ
4 : c2
mX2;
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On the other hand, for n ¼ 2m 1; an odd nonnegative integer number, we can assume l2m 1 ¼
rm 1 ðl 1 Þ ; rm ðl 1 Þ
mX1;
where rm is a polynomial of degree m: Thus, for mX2; (15) becomes c2mþ1 c2m
4ð2m 1Þrm ðl 1 Þ rm 2 ðl 1 Þ ð2m 1Þ2 1 þ 4ð2m 3Þ ¼ l þ c2m 1 rm 1 ðl 1 Þ rm 1 ðl 1 Þ þ 16c2m 1 c2m 2 c2m 3 ;
or, equivalently, rm 1 ðl 1 Þ ð2m 1Þ2 rm ðl Þ ¼ l 1 þ þ 16c2m 1 c2m 2 c2m 3 4ð2m 1Þc2mþ1 c2m c2m 1
!
1
2m 3 rm 2 ðl 1 Þ: ð2m 1Þc2mþ1 c2m
If rm ¼ tm r˜m where r˜m denotes the monic polynomial associated with rm ; then we get ! ð2m 1Þ2 1 1 r˜m ðl Þ ¼ l þ þ 16c2m 1 c2m 2 c2m 3 r˜m 1 ðl 1 Þ c2m 1 16ð2m 3Þ2 c2m 1 c2m 2 r˜m 2 ðl 1 Þ;
mX2;
with initial conditions r˜0 ðl 1 Þ ¼ 1
and
r˜1 ðl 1 Þ ¼ l 1 þ
1 : c1
As a conclusion, fq˜ m g and f˜rm g are sequences of monic orthogonal polynomials.
4. Asymptotics of Qn First, we establish the asymptotic behavior of the sequence fln g which appears in (11). pffiffiffi Proposition 3. For the sequence fln = ng we get the upper bound pffiffiffi 5 ln pffiffiffio ; nX1: n 3 pffiffiffi Furthermore, the sequence fln = ng is convergent and ln 1 lim pffiffiffi ¼ pffiffiffi: n 6 3
n-N
ð16Þ
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Proof. By the extremal property of jjQn jj2S we have Z N Z N 4 2 2 x4 2 Pn ðxÞe dx þ ln P2n 1 ðxÞe x dx: jjQn jjS X N
N
Thus, for nX1; from (11) RN 2 4 pffiffiffi N Pnþ2 ðxÞe x dx ln pffiffiffi ¼ 4l n n jjQn jj2S RN 2 4 pffiffiffi Pnþ2 ðxÞe x dx N R p 4l n R N 2 x4 dx þ ln2 N P2 ðxÞe x4 dx N Pn ðxÞe N n 1 pffiffiffi cnþ2 cnþ1 cn ¼ 4l n : cn þ ln2
ð17Þ
From the recurrence relation (10) we have, for nX2; pffiffiffi n 2 4cn ¼ n 4cn ðcnþ1 þ cn 1 Þon ) cn o ; 2 but simple computations show that the above inequality also holds for n ¼ 1; that is, pffiffiffi n ð18Þ cn o ; nX1: 2 Now, using this inequality in (10) we obtain n 3 pffiffiffiffiffiffiffiffiffiffiffi ¼ cn ðcnþ1 þ cn þ cn 1 Þo n þ 1c n 4 2 pffiffiffiffiffiffi nþ1 and, then cn 4n6ðnþ1Þ ; for nX2; but again simple computations prove that this inequality is true for n ¼ 1; and, therefore, n cn 4 pffiffiffiffiffiffiffiffiffiffiffi; nX1: ð19Þ 6 nþ1 We use relations (18) and (19) in (17) obtaining, for nX2; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðn þ 2Þðn þ 1Þ 5 ln l ðn þ 2Þðn þ 1Þ pffiffiffio p ; nX3; o 1 pffiffiffiffiffiffi þ 2ln 3 2n n 3 nþ1 and straightforward computations in (12) show that this inequality holds for n ¼ 1; 2: Thus, pffiffiffi ln 5 pffiffiffio o1; nX1: ð20Þ n 3 On the other hand, relation (15) can be rewritten as ln ¼ 1
4ncnþ2 cnþ1 2
n l þ cn þ 16cn cn 1 cn 2 4ðn 2Þln 2
;
nX3
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and from here we get for nX3; ln pffiffiffi ¼ n
1 pffiffi 1 4l ncnþ2 cnþ1
þ
1 n3=2 4ðcn cnþ1 cnþ2
þ
16c pffiffin cn 1 cn 2 Þ ncnþ1 cnþ2
n 2 cnþ1 cnþ2
qffiffiffiffiffiffi
:
n 2 plffiffiffiffiffiffi n 2 n n 2
ð21Þ
Denoting
1 1 n3=2 16cn cn 1 cn 2 þ þ pffiffiffi BðnÞ ¼ pffiffiffi ; 4l ncnþ2 cnþ1 4 cn cnþ1 cnþ2 ncnþ1 cnþ2 n 2 CðnÞ ¼ cnþ1 cnþ2
rffiffiffiffiffiffiffiffiffiffiffi n 2 ; n
ð22Þ
ð23Þ
then (21) becomes ln 1 pffiffiffi ¼ : n 2 n BðnÞ CðnÞplffiffiffiffiffiffi
ð24Þ
n 2
Taking into account that in [4] an asymptotic expansion of cn was established, in particular cn 1 lim pffiffiffi ¼ pffiffiffi; n 2 3
ð25Þ
n-N
using (25) we get pffiffiffi 20 3 ; lim BðnÞ ¼ n-N 3
lim CðnÞ ¼ 12:
n-N
ð26Þ
pffiffiffi Therefore, if the sequence fln = ng converges, its limit r must satisfy the equation pffiffiffi pffiffiffi pffiffiffi r ¼ 1=ð20 3=3 12rÞ; i.e., either r ¼ 1=6 3 or r ¼ 3=2: But, from (20), it is p ffiffi ffi pffiffiffi deduced that the limit of fln = ng; if it exists, is 1=6 3: Then, to conclude the proof pffiffiffi of this Proposition it only remains to prove that fln = ng converges. pffiffi pffiffiffi If r ¼ 1=6 3 and y ¼ 35; then, from (24), we have n 2 r2 CðnÞ þ r2 CðnÞj ln j1 rBðnÞ þ rCðnÞplffiffiffiffiffiffi n 2 pffiffiffi r ¼ : n n 2 BðnÞ CðnÞplffiffiffiffiffiffi
ð27Þ
n 2
ln On the other hand, using inequality (20) for the sequence pffiffiffi; we have n n 2 oyCðnÞ and so, for n large enough, CðnÞplffiffiffiffiffiffi n 2 ln 2 BðnÞ CðnÞ pffiffiffiffiffiffiffiffiffiffiffi4BðnÞ yCðnÞ40: n 2
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From here we can give a bound for (27), i.e., for n large enough, n 2 rÞ þ r2 CðnÞj j1 rBðnÞ þ r CðnÞðplffiffiffiffiffiffi ln n 2 pffiffiffi ro n BðnÞ yCðnÞ j1 rBðnÞ þ r2 CðnÞj rCðnÞ ln 2 : p ffiffiffiffiffiffiffiffiffiffiffi þ BðnÞ yCðnÞ p r BðnÞ yCðnÞ n 2 Now, taking into account the limits of the sequences BðnÞ and CðnÞ given in (26), we obtain pffiffiffi ln ln 2 2= 3 pffiffiffi lim suppffiffiffiffiffiffiffiffiffiffiffi r lim suppffiffiffi rp 20 pffiffi 4 5 n-N n n-N n 2 3 ln 2 1 pffiffiffiffiffi lim suppffiffiffiffiffiffiffiffiffiffiffi r; ¼ 10 2 15 n-N n 2 where 1 1 pffiffiffiffiffio ; 10 2 15 2 pffiffiffi and, therefore, we can conclude that the sequence fln = ng is convergent and its limit pffiffiffi is r ¼ 1=ð6 3Þ: & We want to compare the asymptotic behavior of Qn and Pn in the complex plane, more exactly, in C\R: We get the following result: Theorem 2. The asymptotic behavior lim
n-N
Qn ðxÞ 3 ¼ Pn ðxÞ 2
ð28Þ
holds uniformly on compact subsets of C\R: Proof. We consider the orthonormal polynomials pn with respect to the inner RN 4 product N f ðxÞgðxÞe x dx: In [5] Lo´pez and Rakhmanov give the strong asymptotics of pn ; i.e., it holds uniformly on compact subsets of C\R; lim
n-N
pn ðxÞ Dn ðxÞðjðx=xn ÞÞ
nþ1=2
1 ¼ pffiffiffiffiffiffi; 2p
+ function for the weight e x on the segment ½ xn ; xn ; i.e., where Dn ðxÞ is the Szego’s (pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ) x2 x2n xn t4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt ; Dn ðxÞ ¼ exp 2 2 2p xn ðx tÞ xn t pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=4 and jðxÞ ¼ x þ x2 1 is the conformal mapping of C\½ 1; 1 being xn ¼ ð4n 3Þ onto the exterior of the closed unit disk. 4
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Thus, we can deduce that lim
n-N
pn ðxÞ ¼ 1; pnþ2 ðxÞ
uniformly on compact subsets of C\R: Then, for the monic polynomials Pn we get pffiffiffi pffiffiffi nPn ðxÞ ¼ 2 3; ð29Þ lim n-N Pnþ2 ðxÞ uniformly on compact subsets of C\R: Dividing relation (11) by Pn ðxÞ we obtain Qn ðxÞ Pn 2 ðxÞ Qn 2 ðxÞ ¼ 1 ln 2 ; Pn ðxÞ Pn ðxÞ Pn 2 ðxÞ where, using (16) and (29), we have lim ln 2
n-N
Pn 2 ðxÞ 1 ¼ ; Pn ðxÞ 3
uniformly on compact subsets of C\R: Now, standard arguments allow us to conclude that the sequence fQn =Pn g is convergent and its limit is the solution of the equation s ¼ 1 þ s=3; that is, s ¼ 3=2: & From this theorem we deduce that the Sobolev polynomials fQn g have the same asymptotic behavior (up to multiplicative constant factors) as fPn g in C\R: This occurs in other cases when the measures m0 and m1 involved in the Sobolev inner product (1) with N ¼ 1 have unbounded support (see [1,6]) but this is not the case when the measures have compact support (see, for example, [11] or [12]). Three natural questions arise. The first one is why does it occur? The second one is when does it occur? Finally, can we give a more complete description of the asymptotic behavior of the polynomials Qn ? The answer to the first and second questions is yet open for us, but we can obtain better information about the asymptotics of Qn if we scale the variable x in a convenient way, i.e., if we look for the exterior Plancherel– Rotach-type asymptotics for Qn : We have the following result: Theorem 3. The asymptotic behavior qffiffi p ffiffiffi 2 4 3 3j 4 4x Qn ð nxÞ p q ffiffi ffiffi ffi lim ¼ n-N Pn ð 4 nxÞ 3j2 4 3 x þ 1
ð30Þ
4
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi holds uniformly on compact subsets of C\½ 4 4=3; 4 4=3 ; where jðxÞ ¼ x þ x2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi with x2 140 if x41; i.e., the conformal mapping of C\½ 1; 1 onto the exterior of the closed unit disk. Proof. It is well-known (see [16]) that from the three-term recurrence relation xpn ðxÞ ¼ anþ1 pnþ1 ðxÞ þ an pn 1 ðxÞ;
nX1;
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we can obtain asymptotic properties of the orthonormal polynomials pn : Indeed, as an 1 ffiffiffiffiffi; ffiffiffiffiffiffiffiffiffiffi ¼ p lim p 4 4 nþj 12
n-N
for a jAR fixed;
we deduce (see [16]) pffiffiffiffiffiffiffiffiffiffi pn 1 ð 4 n þ j xÞ 1 p ffiffiffiffiffiffiffiffiffiffi lim ¼ qffiffi ; n-N pn ð 4 n þ j xÞ 4 3 j x
j fixed;
4
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Then, for the monic polynomial Pn we have p ffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4 p ffiffiffi Pn 1 ð 4 n þ j xÞ 12 4 pffiffiffiffiffiffiffiffiffiffi ð31Þ lim n ¼ qffiffi ; j fixed; n-N 4 3 Pn ð 4 n þ j xÞ j x 4
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Introducing the change of ffiffi ffi p variable x- 4 nx in (11) and using this relation in a recursive way, we get ½ðn 1Þ=2 X p pffiffiffi ffiffiffi ðnÞ 4 Qn ð n xÞ ¼ ð 1Þj b2j Pn 2j ð 4 n xÞ;
nX3;
j¼0
Q ðnÞ ðnÞ with b0 ¼ 1; b2j ¼ ji¼1 ln 2i ; and ½a denotes the integer part of a: Then, dividing pffiffiffi by Pn ð 4 n xÞ we obtain pffiffiffi Qn ð 4 n xÞ ffiffiffi ¼ p Pn ð 4 n xÞ ¼
½ðn 1Þ=2 X j¼0 ½ðn 1Þ=2 X j¼0
pffiffiffi Pn 2j ð 4 n xÞ ffiffiffi p Pn ð 4 n xÞ Qj pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðnÞ b2j n 2i Pn 2j ð 4 n xÞ j p ffiffi ffi ð 1Þ Qj pffiffiffiffiffiffiffiffiffiffiffiffiffi i¼1 ; Pn ð 4 n xÞ n 2i i¼1 ðnÞ
ð 1Þj b2j
where an empty product is equal to 1. Now, we analyze the asymptotic behavior of the factors in the above sum. If we use (16) in Proposition 3 and (31), then
ðnÞ j Y b2j ln 2i 1 j pffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð 1Þj 6 3 n 2i n 2i i¼1 i¼1
lim ð 1Þj Qj
n-N
pffiffiffi j pffiffiffiffiffiffiffiffiffiffiffiffiffi Y Pn 2j ð 4 n xÞ pffiffiffi n 2i lim n-N Pn ð 4 n xÞ i¼1 0 1j pffiffiffi B 2 3 C ¼ @ qffiffi A ; j fixed: j2 4 34 x
j fixed;
ð32Þ
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pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi This last limit holds uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : Gathering the above limits we get 0 1j p ffiffi ffi 4 1 B C ðnÞ Pn 2j ð n xÞ p ffiffiffi lim ð 1Þj b2j ð33Þ ¼@ qffiffi A ; 4 n-N 4 3 Pn ð n xÞ 2 3j x 4 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 : pffiffiffi On the other hand, the upper bound of the sequence fln = ng obtained in Proposition 3 together with the limit relation (32) allow us to give a uniform bound pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ðnÞ for ð 1Þj b2j Pn 2j ð 4 nxÞ=Pn ð 4 nxÞ on C\½ 4 4=3; 4 4=3 ; that is, for n large enough and 0pjp½ðn 1Þ=2 ; pffiffiffi ð 4 n xÞ ð 1Þj bðnÞ Pn 2jp ffiffiffi pKyj ; 2j Pn ð 4 n xÞ where
pffiffiffi 5 o1 y¼ 3
pffiffiffi pffiffiffi and K is a constant. Therefore, we have a majorant for Qn ð 4 n xÞ=Pn ð 4 n xÞ with ffiffiffiffiffiffiffi ffi p p ffiffiffiffiffiffiffi ffi xAC\½ 4 4=3; 4 4=3 : From Lebesgue’s dominated convergence theorem together with (33) we get 0 1j qffiffi p ffiffiffi 2 4 3 N 3j 4 X 4x Qn ð n xÞ 1 B C p q ffiffi q ffiffi ffiffi ffi lim ¼ ¼ ; @ A n-N Pn ð 4 n xÞ 4 3 j¼0 3j2 3j2 4 34 x þ 1 4x pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi uniformly on compact subsets of C\½ 4 4=3; 4 4=3 and thus the statement of our qffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi theorem follows. Note that 1= 3j2 4 34 x o1 when xAC\½ 4 4=3; 4 4=3 : &
Acknowledgments We thank Professors Jeff Geronimo and Arno Kuijlaars for their suggestions which have improved the presentation of the paper.
References [1] M. Alfaro, J.J. Moreno-Balca´zar, T.E. Pe´rez, M.A. Pin˜ar, M.L. Rezola, Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs, J. Comput. Appl. Math. 133 (2001) 141–150. [2] W.D. Evans, L.L. Littlejohn, F. Marcella´n, C. Markett, A. Ronveaux, On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal. 26 (2) (1995) 446–467.
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