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

2008/12/10 by L. Baratchart, Baratchart, L., S. Kupin +5
Mathematics · #30B70 #41A20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:30B70 #msc:41A20

paper · pdf · doi:10.48550/arxiv.0812.2050

a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extended

arxiv created 2010/02/11 · arxiv updated 2010/02/26

Abstract

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.

Related

• Wall rational functions and Khrushchev's formula for orthogonal rational functions
• Schur analysis and discrete analytic functions: Rational functions and co-isometric realizations
• Schur functions and orthogonal polynomials on the unit circle
• A note on Wall's modification of the Schur algorithm and linear pencils of Jacobi matrices
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics