Sobolev-type orthogonal polynomials on the unit circle

35 pages, no figures.-- MSC2000 codes: 42C05.MR#: MR1891026 (2003e:42037)Zbl#: Zbl 1033.42025This paper deals with polynomials orthogonal with respect to a Sobolev-type inner product $$\langle f,g\rangle =\int_{-\pi}^\pi f(e^{i\theta}) \overline{g(e^{i\theta})} d\mu(e^{i\theta})\, + \, \bold{f}(c)A (\bold{g}(c))^H.$$ where μ is a positive Borel measure supported on [−π,π), A is a nonsingular matrix and 1. We denote f(c)=(f(c),f'(c),\dots,f^{(p)}(c)) and v^H the transposed conjugate of the vector v. We establish the connection of such polynomials with orthogonal polynomials on the unit circle with respect to the measure [see attached full-text file]. Finally, we deduce the relative asymptotics for both families of orthogonal polynomials.The work of the first author (F. Marcellán) was partially supported by D.G.E.S. of Spain under grant PB96-0120-C03-01. The work of the second author (L. Moral) was partially supported by P.A.I. 1997 (Universidad de Zaragoza) CIE-10.Publicad

Approximation theory for weighted Sobolev spaces on curves

17 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.MR#: MR1882649 (2003c:42002)In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete. We also prove the density of the polynomials in these spaces for non-closed compact curves and, finally, we find conditions under which the multiplication operator is bounded on the completion of polynomials. These results have applications to the study of zeroes and asymptotics of Sobolev orthogonal polynomials.Research of V. Álvarez, D. Pestana and J.M. Rodríguez partially supported by a grant from DGI, BFM2000-0206-C04-01, Spain.Publicad

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

Weighted Sobolev spaces on curves

45 pages, no figures.-- MSC1987 codes: 41A10, 46E35, 46G10.MR#: MR1934626 (2003j:46038)Zbl#: Zbl 1019.46026In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials.Research of second (D.P.), third (J.M.R.) and fourth (E.R.) authors was partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad