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Strong asymptotics for the continuous Sobolev orthogonal polynomials on the unit circle. (English) Zbl 0934.42017

Let \(\{Q_n\}\) be a sequence of monic polynomials with \(\deg\widetilde Q_n= n\), which are orthogonal with respect to the Sobolev inner product on the unit circle \(\Gamma= \{\zeta\in\mathbb{C}: |\zeta|= 1\}\), \[ \langle p,q\rangle_{W^1_2(\mu_0, \mu_1)}:= \int_\Gamma p(\zeta)\overline{q(\zeta)} d\mu_0(\zeta)+ \int_\Gamma p'(\zeta)\overline{q'(\zeta)} d\mu_1(\zeta),\tag{1} \] where \(\mu_i\), \(i= 0,1\), are positive Borel measures supported on \(\Gamma\) and \(\mu_1\) is absolutely continuous with respect to the Lebesgue measure, and its Radon-Nikodým derivative is positive and smooth. In this paper, the authors prove that, under convenient conditions on the measures \(\mu_0\) and \(\mu_1\), the sequence \(\widetilde Q_n(z)/z^n\to F_1(z)\), when \(n\to\infty\), uniformly on compact subsets \(K\) of \(\Omega= \{\zeta\in \mathbb{C}: |\zeta|> 1\}\), where \(F_1\) is a Szegö-type function. This is related with recent results of A. Martínez-Finkelshtein [J. Comput. Appl. Math. 99, No. 1-2, 491-510 (1998; Zbl 0933.42013) and “Bernstein-Szegö’s theorem for Sobolev orthogonal polynomials”, Constr. Approx. (to appear)].

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 0933.42013
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References:

[1] Berriochoa, E., Productos escalares con derivadas y modificaciones a través de formas lineales (1997), University of Vigo
[2] Berriochoa, E.; Cachafeiro, A., Lebesgue Sobolev orthogonality on the unit circle, J. Comput. Appl. Math., 96, 27-34 (1998) · Zbl 0933.42011
[3] Foulquié, A.; Marcellán, F., Strong asymptotics for polynomials with respect to a discrete Sobolev inner product on the support of the measure of orthogonality, Methods Appl. Anal., 4, 53-66 (1997) · Zbl 0885.42014
[4] Geronimus, Y. L., Orthogonal Polynomials (1961), Consultants Bureau: Consultants Bureau New York
[5] Golinskii, L. B., Asymptotics for derivatives of orthogonal polynomials, Izv. Akad. Nauk Armenian SSR Mat., 11, 56-81 (1976)
[6] Li, X.; Marcellán, F., On polynomials orthogonal with respect to Sobolev inner product on the unit circle, Pacific J. Math., 175, 127-146 (1996) · Zbl 0897.42013
[7] López, G.; Marcellán, F.; Van Assche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx., 11, 107-137 (1995) · Zbl 0840.42017
[8] Marcellán, F.; Van Assche, W., Relative asymptotics for orthogonal polynomials with a Sobolev inner product, J. Approx. Theory, 72, 193-209 (1993) · Zbl 0771.42014
[9] Martínez-Finkelshtein, A., Asymptotic properties of Sobolev orthogonal polynomials, J. Comp. Appl. Math., 99, 491-510 (1998) · Zbl 0933.42013
[10] A. Martínez-Finkelshtein, Bernstein-Szegő’s theorem for Sobolev orthogonal polynomials, Constr. Approx, to appear.; A. Martínez-Finkelshtein, Bernstein-Szegő’s theorem for Sobolev orthogonal polynomials, Constr. Approx, to appear.
[11] Martínez-Finkelshtein, A.; Moreno-Balcázar, J.; Pijeira-Cabrera, H., Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials, J. Comput. Appl. Math., 81, 211-216 (1997) · Zbl 0895.33003
[12] Martínez-Finkelshtein, A.; Moreno-Balcázar, J.; Pérez, T. E.; Piñar, M. A., Asymptotics of Sobolev Orthogonal Polynomials for Coherent Pairs of Measures, J. Approx. Theory, 92, 280-293 (1998) · Zbl 0898.42006
[13] Martínez-Finkelshtein, A.; Moreno-Balcázar, J., Asymptotics of Sobolev orthogonal polynomials for a Jacobi weight, Methods Appl. Anal., 4, 430-437 (1997) · Zbl 0897.33010
[14] Suetin, P. K., Fundamental properties of polynomials orthogonal on a contour, Russian Math. Surveys, 21, 35-83 (1966) · Zbl 0182.09302
[15] Szegő, G., Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23 (1975), Amer. Math. Soc: Amer. Math. Soc Providence
[16] Widom, H., Extremal polynomials associated with a system of curves in the complex plane, Adv. Math., 3, 127-232 (1969) · Zbl 0183.07503
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