SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART
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1 Georgian Mathematical Journal Volume 4 (27), Number 4, SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics of monic extremal polynomials with respect to the norm L p (σ) are studied. he measure σ is concentrated on the segment [, ] plus a finite set of mass points in a region of the complex plane exterior to the segment [, ]. 2 Mathematics Subject Classification: 42C5, 3E5, 3E. Key words and phrases: Orthogonal polynomials, asymptotic behavior.. Introduction Let σ be a positive Borel measure supported by a compact set E of the complex plane. Denote by n,2 (z) the monic polynomial n,2 (σ, z) = z n + lower degree terms, orthogonal to the measure σ, i.e., ( n,2, z k ) Lp(σ) := n,2 (ξ)ξ k dσ(ξ) =, k =,,..., n. () E It is well known that these polynomials satisfy the extremal property m n,2 (σ) = min Q=z n + Q L 2 (σ), where m n,2 (σ) := ( n,2, n,2 ) /2 L 2 (σ) = n,2 L2 (σ). hus the n-th orthogonal polynomial can be defined as a monic polynomial of degree n with a minimal norm in the Hilbert space L 2 (σ). From this point of view, we can define a large class of monic polynomials n,p (σ, z) = z n + lower degree terms called extremal polynomials that realize a minimal norm in L p (σ), i.e., n,p Lp (σ) = min Q Q=z n + L p (σ) = m n,p (σ) here is vast literature on orthogonal polynomials, but on extremal polynomials is insufficient. A special area of research in this subject has been the study of the asymptotic behavior of n,p (z) when n tends to infinity. Beginning with the results obtained by Geronimus in 952 [], who considered the case where the support E of the measure is a rectifiable Jordan curve, in particular, Widom [8] investigated the case p =. In 987, Lubinsky and Saff proved the asymptotics of m n,p (σ) and n,p outside the segment [, ] under a general condition on the weight function [6]. Another result on zero distributions of ISSN X / $8. / c Heldermann Verlag
2 674 R. KHALDI extremal polynomials on the unit circle was presented by X. Li and K. Pan in [5]. In 992, Kaliaguine [2] obtained the power asymptotics for extremal polynomials when E is a rectifiable Jordan curve plus a finite set of mass points. Recently, Khaldi presented an extension of Kaliaguine s results in [4], where he studied the case of a measure supported on a rectifiable Jordan curve plus an infinite set of mass points. In the special case p = 2 of orthogonal polynomials, Kaliaguine established in [3] the power asymptotics for such a polynomial on an arc plus a finite discrete part, and recently Peherstorfer and Yuditskii [7] solved this problem for a measure supported on the segment [2, 2] plus a denumerable set of mass points which accumulate at the end points of the interval. In this paper, we establish the strong asymptotics of the L p (σ) extremal polynomials { n,p (σ, z)} associated with the measure σ which has a decomposition of the form σ = α + γ, where α is a measure with supp(α) = [, ], absolutely continuous with respect to the Lebesgue measure on the segment E, i.e., + dα(x) = ρ(x)dx, ρ, ρ(x)dx < +, satisfying some extra conditions, and γ is a measure supported on the finite set {z k } l C\[, +], i.e., l γ = A k δ(z z k ); A k >. (2) 2. Preliminaries 2.. Hardy space and the Szegö function. Let E = [, ], Ω = {C\[, ]} { }, G = {w C : w > } { }. he conformal mapping Φ : Ω G is defined by Φ(z) = z + ( z 2, its inverse Ψ(w) = 2 w + ( ) w), and the capacity C(E) = lim Φ(z) =. z z 2 Let ρ be an integrable non negative weight function on E satisfying the Szegő condition Log ρ(x) dx >. (3) x 2 hen we can easily see that the weight function λ defined on the unit circle by { λ(e iθ ρ(ξ)/ Φ (ξ), ξ = Ψ(e iθ ), < θ <, ) = ρ(ξ)/ Φ +(ξ), ξ = Ψ(e iθ ), < θ <, satisfies the usual Szegő condition Log(λ(e iθ )) >.
3 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS 675 hus the Szegő function associated with the unit circle = {t : t = } and the weight function λ is defined by D (w) = exp Log (ρ(cos θ) sin θ ) + weiθ, w <, (4) 2p weiθ and satisfies the following properties: ) D is analytic on the open unit disk U = {w : w < }, D (w), w U, and D () >. 2) D has boundary values almost everywhere on the unit circle such that a.e. for t = e iθ. λ(e iθ ) = ρ(cos θ) sin θ = D(t) p Definition. An analytic function f on Ω belongs to H p (Ω, ρ) if and only if f(ψ(w))/d(/w) H p (G), where H p (G) is the usual Hardy space associated with the exterior G of the unit circle. Any function f H p (Ω, ρ) has boundary values f + and f on both sides of E, and f +, f L p (α). In the Hardy space H p (Ω, ρ) we will define f p H p (Ω,ρ) = where E R = {z Ω : Φ(z) = R}. E f(x) p ρ(x)dx = lim R + R E R p f (z) Φ D (z) p (z) dz, 2.2. Notation and the lemmas. Let p <. We denote by µ (ρ), µ (σ) respectively the extremal values of the problems { } µ (ρ) = inf ϕ p H p (Ω,ρ) : ϕ Hp (Ω, ρ), ϕ ( ) =, (5) µ (σ) = inf { ϕ p H p (Ω,ρ) : ϕ Hp (Ω, ρ), ϕ ( ) =, ϕ (z k ) =, k =, 2,..., l }. (6) We denote by ϕ and ψ the extremal functions of problems (5) and (6). Note that ϕ (z) = D (/Φ(z)) /D () is an extremal function of problem (5) and µ(ρ) = / [D ()] p (see [2]). Lemma (. he extremal functions ϕ and ψ are related by ψ = B(z).ϕ l p and µ(σ) = Φ(z k ) ) µ(ρ), where l Φ(z) Φ(z k ) Φ(z k ) 2 B(z) = (7) Φ(z)Φ(z k ) Φ(z k ) is the Blaschke product. Proof. Proceeding in the same way as in the case of a curve in [2, p. 23], we obtain the following result.
4 676 R. KHALDI Lemma 2. lim sup2 n m n,p (σ) [ µ ( ρ(2 lp ω l p)] /p [µ(σ)] /p, (8) n where ω l (z) = l (z z k ). Proof. he proof is the same as in [2, pp ]. 3. Main results Definition 2. A positive measure α supported on [, ] is said to belong to a class A if it is absolutely continuous with respect to the Lebesgue measure as well as if for each r <, (α ) L r [, ]. First we state the Lubinsky and Saff s result [6]. heorem ([6]). Let α A be a positive measure. We can associate with the measure α, the function D, and the extremal values m n,p (α) and µ(ρ) given by (4), (), and (5). hen the monic extremal polynomials n,p (α, z) have the following asymptotic behavior (n ): () lim 2 n m n,p (α) = [µ(ρ)] /p ; (2) lim n,p (α, z) = {Φ(z)/2} n D (/Φ(z)) [ + χ n (z)] D () where χ n (z) uniformly on the compact subsets of Ω. Now we give the main result of this paper. heorem 2. Let a measure σ = α + γ be such that α A and γ = l A k δ(z z k ). Associate with the measure σ the functions D, B and the extremal values m n,p (σ) and µ(σ) given by (4), (7), (), and (6), respectively. hen the monic extremal polynomials n,p (σ, z) have the following asymptotic behavior as n : () lim 2 n m n,p (σ) = [µ(σ)] /p. (2) n,p (σ, z) = {Φ(z)/2} n B(z) D (/Φ(z)) [ + χ n (z)], D () where χ n (z) uniformly on compact subsets of Ω. Proof. Putting t k = Φ(z k ), ξ = Ψ(t) where t = eiθ, we have B(ξ) = with = l t t k t k 2, k =,..., l. tt k t k Now we consider the integral I n = 2 n t n n,p (Ψ(t)) D(t) ( ) D() + t2n D(t) 2 D()D(t)
5 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS 677 and transform it in a standard way as the following sum I n = 2 n t n n,p (Ψ(t)) 2 D(t) + D() + t2n D(t) 2 D()D(t) 2Re ( ) 2 n t n n,p (Ψ(t)) D(t) D() + t2n D(t) D()D(t). (9) hen applying the Hölder inequality to the first term of (9) we get 2 n t n n,p (Ψ(t)) 2 D(t) 2 n n,p (Ψ(t)) p 2/p D(t) 2/p 2 n n,p (Ψ(t)) p D(t) p ( ) + 2/p 2/p = 2 n n,p (x) p ρ(x)dx 2/p [ ] 2 2 n m n,p (σ). () () /p he second term of(9) we transform as the following sum D() + t2n D(t) 2 D()D(t) = + 2 D() 2 D() + 2Re ( ) t 2n D(t) D() D()D(t). When n tends to, the last term approaches. For the first and the second term we have the estimate 2 D() + 2 D() = 2 D() = l [ ] t k 2 2/p µ(σ) (D()) 2 =. () o estimate the last integral of (9), we transform it as follows: ( 2 n t n n,p (Ψ(t)) t 2n J n = D(t) D() + D(t) ) D()D(t)
6 678 R. KHALDI = = 2 n t n n,p (Ψ(t)) D(t) ( ) t 2n D() + ( ) 2 n t n n,p (Ψ(t)) D(t) D(t) D()D(t) 2 n t n n,p (Ψ(t)) 2 dt D(t) D() it. (2) By using the residue heorem we set J n = l t k 2 (D()) 2 + 2n the last term of (3) can be estimated as l t n k n,p (z k ) D(t k )b (t k ) [ l ( ) t n q ] /q [ l ] /p k D(t k )b (t k ) n,p (z k ) p A k, A /p k l t n k n,p (z k ) D(t k )b (t k ), (3) p + q =. (4) aking into account (4) and Lemma 2, we obtain [ ] 2/p µ(σ) J n + δ n, (5) where δ n as n. Substituting (), () and (5) into (9) we obtain [ ] 2 [ ] 2 n 2/p m n,p (σ) µ(σ) I n 2δ () /p n. (6) Finally, using the previous estimate we get lim inf n 2n m n,p (σ) [µ(σ)] /p. his with Lemma 2 prove the first statement of heorem 2. Now, to prove (2) of heorem 2, first we estimate the integral [ 2 n t n n,p (Ψ(t)) D(t) w ( )] D() + t2n D(t) D()D(t) 2 n t n n,p (Ψ(t)) D(t) dt wt it ( ) D() + t2n D(t) 2 D()D(t) = 2 w I n. (7)
7 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS 679 As an immediate consequence of (6) and () of heorem 2 we get So, [ 2 n t n n,p (Ψ(t)) D(t) lim I n =. (8) n ( )] D() + t2n D(t) D()D(t) dt = o(). (9) wt it On the other hand, we have [ ( )] 2 n t n n,p (Ψ(t)) D(t) D() + t2n D(t) dt D()D(t) wt it dt = χ n (Ψ(t)) wt it t 2n D(t) dt D()D(t) wt it. (2) Applying the Cauchy formula to the first term in (2), we can see that dt χ n (Ψ(t)) wt it = χ n(z), z = Ψ(w) Ω. (2) Since the last term in (2) approaches as n, from (9), (2) and (2) we obtain the second statement of heorem 2. Remark. In this work, we have studied the strong asymptotics of L p extremal polynomials under the Szegö condition for p 2 and the case of a measure with finite discrete part. he cases where < p < 2 and the measure is concentrated on an infinite part, still remain open problems. Acknowledgement he author expresses his sincere gratitude to Prof. F. Marcellan for guidance and valuable suggestions. References. Ya. L. Geronimus, On some extremal problems in the space L (p) σ. (Russian) Mat. Sbornik (N.S.) 3(73)(952), V. A. Kaliaguine, On asymptotics of L p extremal polynomials on a complex curve ( < p < ). J. Approx. heory 74(993), No. 2, V. A. Kaliaguine, A note on the asymptotics of orthogonal polynomials on a complex arc: the case of a measure with a discrete part. J. Approx. heory 8(995), No., R. Khaldi, Strong asymptotics for L p extremal polynomials off a complex curve. J. Appl. Math. 24, No. 5, X. Li and K. Pan, Asymptotics for L p extremal polynomials on the unit circle. J. Approx. heory 67(99), No. 3, D. S. Lubinsky and E. B. Saff, Strong asymptotics for L p extremal polynomials ( < p ) associated with weights on [, ]. Approximation theory, ampa (ampa, Fla., ), 83 4, Lecture Notes in Math., 287, Springer, Berlin, 987.
8 68 R. KHALDI 7. F. Peherstorfer and P. Yuditskii, Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Amer. Math. Soc. 29(2), No., (electronic). 8. H. Widom, Extremal polynomials associated with a system of curves in the complex plane. Advances in Math. 3(969), (969). Author s address: Department of Mathematics University of Annaba B.P.2, 23 Annaba Algeria rkhadi@yahoo.fr (Received )
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