Extremal Polynomials with Varying Measures

Transcription

1 International Mathematical Forum, 2, 2007, no. 39, Extremal Polynomials with Varying Measures Rabah Khaldi Deartment of Mathematics, Annaba University B.P. 12, Annaba, Algeria Fateh Aggoune Guelam University, Faculty of Sciences B.P 401, Guelma, Algeria Abstract We investigate the strong asymtotics for L -extremal olynomials with resect to varying measures on a rectifiable Jordan curve erturbed by a finite Blaschke sequence of oint masses outside the curve Mathematics Subject Classification: 42C05, 30E10 Keywords: Extremal Polynomials, Varying measures 1 Introduction Let σ be a finite ositive Borel measure on a comact set of the comlex lane whose suort contains an infinite set of oints. We denote by T n,2 (z) = z n +..., the monic olynomial of degree n orthogonal with resect to the measure σ. One of the most useful tools in the study of orthogonal olynomials is the fact that they solve the following extremal roblem: minimize L 2 (σ) norm for all monic olynonials of degree n i.e. T n,2 2 L 2 (σ) := min z n + Q 2 Q P L 2 (σ) = m n,2 (σ) n 1 where P n denotes the set of olynomials of degree at most n. This characterization of orthogonal olynomials ermits us to define a larger class of olynomials called extremal olynomials that solve the extremal roblem:

2 1928 R. Khaldi and F. Aggoune T n, L (σ) := min z n + Q 2 Q P L (σ) = m n, (σ). n 1 One of the major areas of research in the study of extremal olynomials is to investigate the strong asymtotics behavior of T n, (z) asn. Other commonly used names are ower asymtotic, Szegö asymtotic, or full exterior asymtotic. The connection of orthogonal olynomials with other branches of mathematics is actually remarkable. We can mention some of them like, moment roblems, otential theory, aroximation theory, secial functions, scattering theory, differential equations, dynamic systems, oerator theory, number theory, random matrices, statistics and robability theory, stochastic rocesses, electrostatics... There exists an extensive literature on orthogonal olynomials but not enough on extremal olynomials, beginning by Geronimus results in 1952 [15, then the fundamental aer of Widom in 1969 [14],.Kaliaguine in [8], obtained the ower asymtotic for extremal olynomials when E is a rectifiable Jordan curve lus a finite set of mass oints. Recently, one of the authors studied in [10,11] the case of a measure suorted on a rectifiable Jordan curve and circle erturbed by an arbirary Blaschke sequence of oint masses. In the last years extremal olynomials with resect to varying measures have layed a major role in several roblems of aroximation theory and growing attention has been aid to the study of their different tyes of asymtotics behavior. This is not accedently, they become a owerful tool in solving roblems where a fixed measure and orthogonality are involved. We mention that the first erson that worked with varying measures was Gonchar [6], then Loez and Gonchar [7] introduced them for studying general results of convergence of multioint Padé aroximants for Cauchy transformation. Moreover, Ysen and Loez [4], studied the strong asymtotic of orthogonal olynomials with resect to varying measures in the case of the unit circle and the real line, then they obtain certain relations for the olynomials involved in the contruction of Hermite-Padé aroximation of a Nikishin system of functions. We find some resuts concerning the study of the strong asymtotics for extremal olynomials with varying measures in various degree of generality in [1,3,13]. In this work we try to carry over some of the main ideas of [3] for L - extremal olynomials with resect to varying measures on a rectifiable Jordan curve erturbed by a finite Blaschke sequence of oint masses outside the curve, where the situation turns out to be much more difficult. Before setting the main results, let us introduce some notations. Let E be an analytic Jordan curve in the comlex lane, we denote by B the bounded comonent of the comlement of E and let ϕ be the confomal maing which mas the oen unit disc D = {z : z < 1} onto B such that

3 Extremal olynomials with varying measures 1929 ϕ(0) = ω B and ϕ (0) > 0. From Caratheodory Theorem the function ϕ an its inverse ν = ϕ 1 have a continuous extention to the unit circle and on E, resectively. Let σ(s) be a ositive measure on [0,l[, where l is the length of E and σ (s) the Radon-Nikodym derivate of σ(s) with resect to the Lebesgue measure dξ on E, we suose that it satisfies the Szegő s condition: E (log σ (s)) ν (ξ) dξ >, (1) then we see that for the measure μ defined on the unit circle by yields μ(a) =σ(ξ 1 (ν 1 (A))) = σ((ν ξ) 1 (A)), σ (s) dξ = σ (s) ϕ (θ) dθ = μ (θ)dθ. Thus the measure μ satisfies on the unit circle the usual Szegő s condition: 2π 0 log(μ (θ))dθ >. Therefore, from the roerties of the Szegő function D (μ,.) associated with the unit circle and the weight function μ (θ) (see [12]): { 1 2π D (μ, w) = ex 2π 0 } Log(μ (θ)) 1+we iθ dθ,w D (2) 1 we iθ we can easily deduce the following roerties for the Szegő function Δ (σ, z) = D (μ, ν(z)) associated with the curve E and the weight function σ (θ): 1) Δ (σ,.) is analytic on B, Δ (σ, z) 0onB, and Δ (σ, ω) > 0. 2) ν (ξ) Δ (σ, ξ) = σ (s), almost everywhere on E. 2 Auxiliary results We need to introduce the following auxiliary results. But first we set some notations. H (μ) is defined as the L (μ) closure of the olynomial in e iθ and f H (μ) = f ( e iθ ) dμ(θ). We denote by H = H (m), where m = dθ 2π is the normalized Lebesgue measure on [0, 2π[. Let K (μ, z) be the function such that K (μ, z) = D (μ, 0) D (μ, z) if z (S a {z : z < 1} and K (μ, z) = 0 for z S s,

4 1930 R. Khaldi and F. Aggoune where S a and S s are a disjoint decomosition of the unite circle such that μ and μ s live on these sets resectively. Hereafter, μ s denotes the singular arts of μ with resect to the Lebesgue measure. L 2 s(μ) ={f L 2 (μ) :f =0,μ a.e}. Theorem 1 ([2]). If f H (μ), then there existe unique functions f,f s such that f = K f + fs, f H, and f s L 2 s(μ). Now we state Keldysh Theorem: Theorem 2([9]). Assume that the measure μ satisfies the Szegő condition and {f n } H (μ), 0 <<, such that i) lim fn (0) = 1; ii) lim f n H (μ) = D (μ, 0) Then a) lim fn (z) = 1 holds uniformly on each comact subset of D. b) lim f n K (μ, z) H (μ) =0. Notice that H (σ ) denotes the sace of analytic functions f on.b, such that (f ϕ)(δ (σ,.) ϕ) H. For 1,H (σ ) is a Banach sace, with the norm: f H (σ ) = lim 1 f ( ( )) ϕ re iθ Δ (σ, ϕ ( re iθ) ) dθ. r 1 2πr For 0 <<1, H (σ ) is a quasi-banach sace. Lemma 1([8]). If f H (σ ) then for every comact set K B there is a constant C K such that: 3 Main results su { f (z) : z K} C K f H (σ ). We conserve the same notations as above. Set dα n = σ n + γ = dσ Y n + γ, where the oint measure γ = N A k δ zk is suorted on {z k } N k=1 C\B with k=1 masses A k > 0,k =1,...N and {Y n } n=1 is a sequence of olynomials such that, for each n, Y n has degree n, all its zeros w n,i, i =1,..., n lie in the unbounded comonent of the comlement of the curve Φ(z) = R>1, where Φ is a function which mas the exterior of E onto the exterior of the unite circle G = {w : w > 1}. We will study the asymtotic behavior of olynomials denoted by Tn, N (z) that solve the next extremal roblem

5 Extremal olynomials with varying measures 1931 m n, (α n ) = min P n(ω)=1 { 1 2π E P n (ξ) Y n (ξ) dσ(s)+ } 1/ N A k P n (z k ). (3) Denote by T n, (z) the extremal solution of the following roblem m n, (σ n ) = { 1 min P n(ω)=1 2π E P n (ξ) Y n (ξ) k=1 dσ(s)} 1/. (4) Theorem 3. Let 0 <<, then for the extremal value m n, (α n ) of the roblem (3) we have: lim m n, (α n )=Δ (σ, ω) =D (μ, 0). Proof. The extremal roerty of T n, (z) imlies that { 1 m n, (σ n ) 2π E Tn, N (ξ) Y n (ξ) On the other hand it is roved in [3] that dσ(s)} 1/ m n (α n ). (5) Using (5) and (6) we get lim n + m n, (α n )=Δ (σ, ω) =D (μ, 0). (6) lim inf m n, (α n ) lim inf m n, (σ n )=Δ (σ, ω) =D (μ, 0). (7) n + n + Now let q N (z) be the olynomial whose zeros are z 1,..., z N and q N (α) =1 that is q N (z) = N (z z k )/ N (ω z k ), then the extremal roerty of Tn N (z) imlies that k=1 k=1 (m n, (α n )) min = 1 2π P n N (ω)=1 E { 1 2π E T n N, (ξ) Y n (ξ) P n N (ξ) Y n (ξ) q N (z) dσ(s), } q N (z) dσ(s)

6 1932 R. Khaldi and F. Aggoune where the olynomials T n N, (z) are extremal with resect to the measure dη(s) =q N (z)dσ(s). it is rove in [3, th ] that So, { 1 lim 2π E T n N, (ξ) Y n (ξ) dη(s)} 1/ =Δ (dη, ω). From the roerties of the Szegő function, we obtains that Δ (dη, ω) =D (μ, 0), lim sum n, (α n ) D (μ, 0). (8) l + Inequalities (7) and (8) rove Theorem 3. Theorem 4. For 0 <<, the extremal olynomials Tn, N (z) have the following asymtotics behavior ( n ): T N n, (i) lim ψ =0 Y n H 2 (σ ) (ii) Tn, N (z) =[ψ (z)+ε n (z)], ε n (z) 0 uniformly on the comact subsets of B. Where the function ψ (z) = Δ (σ, ω) Δ (σ, z) Proof. Puting h n (w) = T n, N (ϕ(w)) D (μ, w) = T n, N (z)δ (σ, z), with z = ϕ(w). Y n (ϕ(w)) D (μ, 0) Y n (z)δ (σ, ω) Since the sequence of functions {h n } satisfy the hyothesis of Theorem 2, that is (i) h n (0) = T n, N (ω) D (μ, 0) Y n (ω) D (μ, 0) =1, and from Theorem 3 we obtain T N n, (ii) lim = lim Y n m n, (α n )=D (μ, 0). H 2 (σ ) Then it yields and Tn, N lim D (μ, 0) Y n D (μ, e iθ ) H 2 (σ ) = 0 (9) lim h n(w) =1, (10)

7 Extremal olynomials with varying measures 1933 uniformly on each comact subset of D. From (9) and the fact that ψ (ξ) = Δ (σ, ω) Δ (σ, ξ) = D (μ, 0) D (μ, e iθ ),ξ = ϕ(eiθ )we get the affirmation (i) of the Theorem. Now we will resent two roofs for the affirmation (ii) of the Theorem. First roof of (ii): Using (10) and the exression of h n (w) we obtain Tn, N (z) lim Y n (z) = Δ (σ, ω) Δ (σ, z). (11) uniformly on each comact subset of B. Second roof of (ii): By alying Lemma 1 for the function T N n, which belongs to H 2 (σ ), then for all comact K B, we have Y n ψ su z K T n, (z) Y n (z) ψ (z) C K T N n, Y n ψ 2 H 2 (σ ) 0. References [1] M. P. Alfaro, M. Bello Hernandez, J. M. Montaner, J. L. Varona, Some asymtotic roerties for orthogonal olynomials with resect to varying measures. J. Arox. Theory, 135 (2005), [2] M. Bello Hernandez, F. Marcellan, J. Minguez Ceniceros, Pseudo-uniform convexity in H and somme extremal roblems on Sobolev saces, Comlex Variables Theory and Alication, 48 (2003), [3] M. Bello Hernandez, J. Minguez Ceniceros, Asymtotics for extremal olynomials with varying measures, Electronic Transactions on Numerical Analysis, 19 (2005) [4] B.de la Calle Ysern, G. Loez Lagamasino, Strong asymtotics of orthogonal olynomials with varying measures and Hermite-Padé aroximants, Journ.of Comut. al Math. 99 (1998) [5] L. Geronimus, On some extremal roblems in L () σ 31(1952), saces, Math. Sbornik [6] A. A. Gonchar, On the convergence of Pade aroximants for some classes of meromorhic functions, Mat. Sb. 97(139) (1975),

8 1934 R. Khaldi and F. Aggoune [7] A. A. Gonchar and G. Loez, On Markov s theorem for multi-oint Pade aroximations, Mat. Sb. 105(147) (1978), [8] V.A., Kaliaguine, On asymtotics of L extremal olynomials on a comlex curve (0 << ), J. Arox. Theory, 74 (1993), [9] M. V. Keldysh, Selected aers, Academic ress. Moscow, [in Russian] [10] R. Khaldi, Strong asymtotics for L extremal olynomials off a comlex curve, J. Al. Math (2004), no 5, [11] R. Khaldi, Strong asymtotics of L -extremal olynomials off the unit circle. Demonstratio Math. Vol. 38, No.3, (2005), [12] G. Szegő, Orthogonal lynomials, 4th ed. Amer. Math. Soc. Colloquium Publ. Vol 23. Amer. Math. Soc. Providence, RI, [13] V. Totik, Weighted Aroximation with Varying Weight, Lecture Notes in Math.,Vol. 1569, Sringer-Verlag, Berlin, [14] H. Widom, Extremal olynomials associated with a system of curves in the comlex lane, Adv. in Math. 3 (1969), Received: November 9, 2006