Berriochoa, E.; Cachafeiro, A.; Garcia-Amor, J. Asymptotics on the support for Sobolev orthogonal polynomials on a bounded interval. (English) Zbl 1080.42023 Comput. Math. Appl. 50, No. 3-4, 381-391 (2005). Summary: In the present paper we give sufficient conditions on the measures of orthogonality in order to establish the asymptotic behavior on the support of the measures for the Sobolev orthogonal polynomials with respect to a Sobolev inner product \[ \bigl\langle f(x),g(x)\bigr \rangle_{s_\mu}=\int^1_{-1}f(x)g(x) d\mu_0(x)+\int^1_{-1}f'(x)g'(x)d \mu_1(x), \] with \(\mu_0\) and \(\mu_1\) two finite positive Borel measures on \([-1,1]\). In this situation we prove that the monic Sobolev orthogonal polynomials behave, inside the support, like the monic orthogonal polynomials with respect to a rational modification of the measure \(\mu_1\). We remark that we obtain the results using the connection between Sobolev orthogonal Laurent polynomials on the circle and Sobolev orthogonal polynomials on the interval and using the asymptotic theory of Sobolev orthogonal Laurent polynomials on the circle, that we develop in this paper, and which is important by itself. MSC: 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:orthogonal polynomials; Sobolev inner products; Laurent polynomials; measures on the real line; measures on the unit circle; Szegö function; Carathéodory function PDFBibTeX XMLCite \textit{E. Berriochoa} et al., Comput. Math. Appl. 50, No. 3--4, 381--391 (2005; Zbl 1080.42023) Full Text: DOI References: [1] Berriochoa, E.; Cachafeiro, A.; García-Amor, J., Connection between interval and unit circle for Sobolev orthogonal polynomials. Strong asymptotics on the real line, Acta Appl. Math., 86, 3, 221-236 (2005) · Zbl 1076.42016 [2] Berriochoa, E.; Cachafeiro, A., Strong asymptotics inside the unit disk for Sobolev orthogonal polynomials, Computers Math. Applic., 44, 1/2, 253-261 (2002) · Zbl 1229.33021 [3] Berriochoa, E.; Cachafeiro, A., A necessary condition for the extension of Szegő’s asymptotics inside the disk in the Sobolev case, J. Comput. Appl. Math., 153, 73-78 (2003) · Zbl 1014.42018 [4] Berriochoa, E.; Cachafeiro, A.; García-Amor, J., Asymptotic properties for Chebyshev-Sobolev orthogonal polynomials, J. Comput. Appl. Math., 178, 63-74 (2005) · Zbl 1060.42016 [5] Geronimus, L. Y., Orthogonal Polynomials (1961), Consultants Bureau: Consultants Bureau New York [6] Simon, B., Orthogonal polynomials on the unit circle, (Amer. Math. Soc. Colloq. Pub., 54 (2005)) · Zbl 1127.47037 [7] Szegő, G., Orthogonal Polynomials, (Amer. Math. Soc. Colloq. Pub., Volume 23 (1975)), Amer. Math. Soc. Providence, RI · JFM 65.0278.03 [8] Martínez-Finkelshtein, A., Analytic aspects of Sobolev orthogonal polynomials revisited, J. Comput. Appl. Math., 127, 1/2, 255-266 (2001) · Zbl 0971.33004 [9] Kim, D. H.; Kwon, K. H.; Marcellán, F.; Yoon, G. J., Zeros of Jacobi-Sobolev orthogonal polynomials, Int. Math. J., 4, 5, 413-422 (2003) · Zbl 1232.33015 [10] Meijer, H. G.; de Bruin, M. G., Zeros of Sobolev orthogonal polynomials following from coherent pair, J. Comput. Appl. Math., 139, 2, 253-274 (2002) · Zbl 1005.42015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.