Electronic Transactions on Numerical Analysis. Volume 9,. 29-36, 25. Coyright 25,. ISSN 68-963. ETNA ASYMPTOTICS FOR EXTREMAL POLYNOMIALS WITH VARYING MEASURES M. BELLO HERNÁNDEZ AND J. MíNGUEZ CENICEROS Abstract. In this aer, we give strong asymtotics of extremal olynomials with resect to varying measures of the form dσ n = dσ, where σ is a ositive measure on a closed analytic Jordan curve C, and { is a sequence of olynomials such that for each n, has exactly degree n and all its zeros (α n,i ), i =, 2,..., lie in the exterior of C. Key words. Rational Aroximation, Orthogonal Polynomials, Varying Measures. AMS subject classifications. 3E, 4A2, 42C5.. Introduction. The first erson that worked with varying measures was Gonchar in 978, in [2]. He used them for studying the seed of the best rational aroximation on an interval E R of Cauchy transforms with finite ositive measures and whose suorts lie in an interval F R, with E and F verifying E F =. The varying measures have also been used for other works. Gonchar and Lóez-Lagomasino in [3] studied general results of convergence of multioint Padé aroximants for Cauchy transform. Moreover, in [5] Lóez-Lagomasino resented the varying orthogonal olynomials in a way that joins orthogonal olynomials resect to measures with bounded and unbounded suort and he roved results for orthogonal olynomials resect to measures with unbounded suort using orthogonal olynomials resect to varying measures on z =. In [6] he gave a Szegő theorem for olynomials orthogonal with resect to varying measures. We want to extend this result for L -extremal olynomials with resect to varying measures on closed analytic Jordan curves C. This main theorem will be given in this section and roved in Section 3. In Section 2 we set some auxiliary results for roving the main theorem. Finally, in Section 4 we give a theorem of density. We begin by introducing some notations. Let C be a closed analytic Jordan curve with length l in the z-lane in whose interior α lies. Let σ(s) be a ositive measure on [, l]. We denote by L (C, σ) the sace of measurable and comlex functions on C, such that f,σ = { / f(ζ) dσ(s) <, ζ = ζ(s), C with ζ = ζ(s) a arametrization of C. When C = {z : z =, as usual, we write L (σ). Let B denote the interior of C. Let x = ϕ(z) = α + z + b 2 z 2 +, z <, α B be the conformal transform which mas D = {z : z < onto B, such that α = ϕ() and ϕ () >. From Caratheodory theorem ϕ can be continuously extended to a function which is an one-to-one maing from { z = onto C. Let z = γ(x) denote the inverse Received October 3, 22. Acceted for ublication May, 23. Communicated by F. Marcellán. This work has been suorted by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Sain under grant BFM2-26-C4-3. Dto. de Matemáticas y Comutación, Universidad de La Rioja, Edificio J. L. Vives, C/ Luis de Ulloa, s/n, 264 Logroño, Sain. E-mail: mbello@dmc.unirioja.es. Dto. de Matemáticas y Comutación, Universidad de La Rioja, Edificio J. L. Vives, C/ Luis de Ulloa, s/n, 264 Logroño, Sain. E-mail: judit.minguez@dmc.unirioja.es. 29
3 Asymtotics for extremal olynomials with varying measures function of ϕ. Then the measure σ induces an image measure µ on z = by µ(e) = σ(ζ (γ (E))) = σ((γ ζ) (E)), thus. σ (s) dζ = σ (s) ϕ (e iθ ) dθ = µ (θ)dθ, with σ and µ absolutely continuous arts of σ and µ, resectively. Let K be the unbounded comonent of the comlement of C and z = φ(x) a function which mas K onto E = {z : z > so that the oints at infinity corresond to each other and φ ( ) >. Let C R denote generically the curve φ(x) = R > in K. We define the varying measure (.) dσ n = dσ, where {, n N is a sequence of olynomials such that for each n, has exactly degree n, all its zeros (α n,i ), i =, 2,..., lie in the unbounded comonent of the comlement of a C A, A >, and (α) =. We want to study the asymtotic behavior of olynomials that solve the next extremal roblem { ρ n, = inf Q n,σn = inf Q n (ζ) / (.2) Q n(α)= Q n(α)= C (ζ) dσ(s). We denote by {P n, a sequence of the extremal olynomials, i. e., (.3) P n,,σn = min Q n(α)= Q n,σn. H (C, σ) is defined as the L (C, σ) closure of the olynomials in the variable ζ C. L s(c, σ) = {f L (C, σ) : f =, σ a.e. and L a(c, σ) = {f L (C, σ) : f =, σ s a.e., with σ s the singular art of σ. Similarly, we define Hs (C, σ) and H a (C, σ). Furthermore, we denote by H the classical Hardy sace in { z < and H (µ) the L (µ) closure of the olynomials in e iθ. We suose that σ satisfies the Szegő condition, i. e., log σ (s) γ (ζ) dζ > and this is the same as, log µ L. We denote by D (µ, z) the Szegő function and { ζ + z D (µ, z) = ex ζ z log µ (θ)dθ, ζ = e iθ, z D C (.4) (σ, x) = D (µ, γ(x)). The function (σ, x) satisfies the following roerties: ) (σ, x) is regular in B, more recisely, (σ, x) H (C, σ). 2) (σ, x) in B, and (σ, α) = D (µ, ) >, 3) γ (ζ) (σ, ζ) = σ (s) almost everywhere in C, with ζ C. The main result of this aer that will be roved in Section 3 is the following theorem: THEOREM.. For < <, the following statements are equivalent i) σ satisfies the Szegő condition.
ii) The next it exists and is ositive M. Bello Hernández and J. Mínguez Ceniceros 3 (.5) ρ n, >. iii) There exists a function S(x) regular in B with S(α) = and S,σ <, such that P n, (x) (x) S(x) (.6) =.,σ iv) There exists a function T (x) regular in B such that (.7) P n,(x) (x) Pn,,σ = T (x) holds uniformly on each comact subset of B. Moreover, if i) holds ρ n, = (σ, α), S(x) = (σ, α) (σ, x), and T (x) = (σ, x). 2. Auxiliary Results. Before we can rove the theorems in the following sections, we need to establish several auxiliary results. Let F be a closed ited oint set and let f(x) be a function continuous on F. Let F n be the set of functions of form (2.) π n (x) = b n,x n + b n, x n + + b n,n (x α n, )(x α n,2 ) (x α n,n ). Pick r n (f) F n such that it is the best aroximation to f(x) on F in the sense of Tchebycheff, i. e. f r n (f) = min{ f π n : π n F n with the suremun norm on F. THEOREM 2.. (see [8],. 253). Let C be a closed analytic Jordan curve and let the oints {α n,k, k =, 2,..., n; n =,, 2,... be given with no it oints interior to C A. Let f(x) be analytic on and within C T, then there exists a sequence r n (x) of functions of form (2.) such that r n(x) = f(x), uniformly for x on each closed subset of C R, where R = A2 T + T + 2A 2AT + A 2 +. The Keldysh theorem is another auxiliary theorem to rove Theorem.. This theorem can be seen in [4]. Before introducing it we need the next result which can be found in []. If f H (µ), then there exist unique functions f, f s such that (2.2) f = K f + fs, f H, and f s L s (µ), with (2.3) K (µ, z) = { D(µ,) D, (µ,z) if z (S a {z : z < ),, if z S s,
32 Asymtotics for extremal olynomials with varying measures where S a and S s are a disjoint decomosition of the unit circle such that µ and µ s live on these sets resectively THEOREM 2.2. Assume that µ satisfies the Szegő condition and {f n H (µ), < <, such that i) ii) f n () = ; f n,µ = D (µ, ). Then a) f n (z) = holds uniformly on each comact subset of D. b) f n K (µ, z),µ =. An extension of this theorem is given in []. 3. Proof of Theorem.. Before roving Theorem., we are going to rove an intermediate result. THEOREM 3.. For < < P n, = (σ, α).,σ where relaces (σ, α) if σ does not satisfy the Szegő condition. Proof. Let µ be the image measure of σ on z = by γ ζ. From Szegő, [7]. 297, we know that if T n,2 (z), are the extremal olynomials such that then T n,2 2,µ = min{ Q n 2,µ : Q n monic of degree n, (3.) T n,2 2 2,µ = T n,2 2 2,µ = D 2 (µ, ) 2, with D 2 (µ, ) = if log µ (θ) is not integrable and T n,2 (z) = zn Tn,2 ( z ). Let δ n(x) = (Tn,2 (γ(x)))2/ which is analytic in B since the zeros of Tn,2 lie in { z >, then from { Rmn Theorem 2., there exists a sequence Y mn such that su R mn (ζ) Y mn (ζ) δ n(ζ) =, ζ C and the convergence is uniform in B. In articular, there is convergence in x = α and as δ n (α) = (T n,2 ())2/ = = Y mn (α) we have R mn (α) =. Hence C R mn (ζ) Y mn (ζ) dσ(s) = δ n (ζ) dσ(s) = C therefore (3.2) su T n,2(e iθ ) 2 dµ(θ) = D 2 (µ, ) 2 = D (µ, ), P n, D (µ, γ(α)) = (σ, α).,σ
On the other hand, using Jensen inequality P n, (ζ) C (ζ) dσ(s) = { ex M. Bello Hernández and J. Mínguez Ceniceros 33 log P n, (ϕ(e iθ )) (ϕ(e iθ )) P n, (ϕ(e iθ )) (ϕ(e iθ )) dµ(θ) P n, (ϕ(e iθ )) (ϕ(e iθ )) µ (θ)dθ { dθ ex P n, (ϕ()) (ϕ()) D (µ, ) = D (µ, ). Hence, with this and (3.2) we obtain P n, = D (µ, γ(α)).,σ log µ (θ)dθ Proof of Theorem.: Proof. i) ii). It is done in Theorem 3.. i) iii). We consider the function (3.3) h n (z) = P n,(ϕ(z))d (µ, z) (ϕ(z))d (µ, ), that is regular in D and h n () =. From i) ii) and D (µ, e iθ ) = µ (θ), we have (3.4) { then, alying Theorem 2.2 { hence and then { { h n (e iθ ) dθ =, h n (e iθ ) dθ =, P n, (ϕ(e iθ ))D (µ, e iθ ) (ϕ(e iθ ))D (µ, ) P n, (ϕ(e iθ )) (ϕ(e iθ )) D (µ, ) D (µ, e iθ ) Therefore, using (3.4) and Theorem 3., we have { P n, (ϕ(e iθ )) (ϕ(e iθ )) D (µ, ) D (µ, e iθ ) dθ = µ (θ)dθ =. dµ(θ) =,
34 Asymtotics for extremal olynomials with varying measures and this is the same as iii), where S(x) = (σ, α) (σ, x). iii) i). It follows from the relation (3.5) inf P n, ψ =, where ψ(ζ) is such that ψ. In fact, from (3.5) it follows that there exists a subsequence {n ν such that P nν, ν ψ ν =. If i) does not hold, from ii) P nν, ν =, ν and we obtain ψ =, that it is a contradiction. iii) iv). The sequence of functions {h n as in (3.3) satisfies the hyothesis of Theorem 2.2, hence h n(z) =, holds uniformly on each comact subset of B. Now, since i) is equivalent to iii), from Theorem 3. we have P n,(x) (x) Pn,(x) (x) iv) i). From iv) and Theorem 3. we have P n,(α) (α) Pn,(x) (x) but this is true if and only if i) holds. = = T (α) = D (µ, γ(x)). (σ, α) <, 4. Density Theorem. In this section we give a density theorem that may be seen as an alication of the main theorem. We introduce the notation: R n,k = { h : degree h n k. THEOREM 4.. Let σ a be a measure in C absolutely continuous with resect to Lebesgue measure, and that satisfies the Szegő condition, then the following statements are equivalent: i) For each j Z + P n,n j,,σa = (σ a, α), where P n,n j, denotes an extremal olynomial, with P n,n j, (α) =, i.e. { P n,n j, Q n j = min : Q n j Π n j, Q n j (α) =.,σa,σa
M. Bello Hernández and J. Mínguez Ceniceros 35 ii) For each k Z +, R n,k is dense in H (σ a ). Proof. ii) = i) Here we use the same technique as in the roof of Theorem 3.. From Szegő, [7], we know that the result is true for T n k,2, then T n k,2 2 = D 2 (µ, ) 2. Let δ n k (x) = (Tn k,2 (γ(x)))2/ which is analytic in B since the zeros of Tn k,2 lie in { z >, then from ii) there exists a sequence of olynomials {R mn k of degree m n k such that su R mn k(ζ) δ n k (ζ) Y mn (ζ) = ζ C and the convergence is uniform in B. Then a) R m n k(α) = ; b) R mn k Y mn = (σ a, α).,σ a Given Λ N an index sequence, from ii) we observe that the sequence {m n can be chosen in Λ, so i) follows from a) and b). i) = ii). Set i, j Z +, using i) and Theorem 2.2 we have P n,n (i+j), (ζ) (ζ) (σ a, α) (σ a, ζ), in L (C, σ a ), thus ζ i P n,n (i+j), (ζ) (ζ) ζ i (σ a, α) (σ a, ζ) in H (C, σ a ). Because of H (C, σ a ) = H (C) (σa,α) (σ a, ) and H (C) is the closure of the olynomials in L (C), R n,j satisfies ii). REFERENCES [] M. BELLO HERNÁNDEZ, F. MARCELLÁN, J. MÍNGUEZ CENICEROS, Pseudouniform convexity in H and some extremal roblems on Sobolev saces, Comlex Variables Theory and Alication, 48 (23),. 429-44. [2] A. A. GONCHAR, On the seed of rational aroximation of some analytic functions, Math. USSR Sbornik., 34 (978), No. 2,. 47-63. [3] A. A. GONCHAR, G. LÓPEZ LAGOMASINO, On Markov s theorem for multioint Padé aroximants, Math. USSR Sbornik., 34 (978), No. 4,. 52-524. [4] M. V. KELDYSH, Selected aers, Academic Press, Moscow, 985. [In Russian] [5] G. LÓPEZ LAGOMASINO, Relative Asymtotics for olynomials orthogonal on the real axis, Math. USSR Sbornik., 65 (988), No. 2,. 55-529. [6] G. LÓPEZ, Szegő theorem for olynomials orthogonal with resect to varying measures, in Orthogonal olynomials and their alications (Segovia, 986),. 255 26, Lecture Notes in Math., 329, Sringer, Berlin, 988.
36 Asymtotics for extremal olynomials with varying measures [7] G. SZEGŐ, Orthogonal Polynomials, 4th ed., American Math. Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, Rhode Island, 975. [8] J. L. WALSH, Interolation and Aroximation by Rational Functions in the Comlex Domain, 3th ed., American Math. Society Colloquium Publications, Vol. XX, Amer. Math. Society, Providence 6, Rhode Island, 96.