Orthogonal rational functions and quadrature on an interval

AbstractRational functions with real poles and poles in the complex lower half-plane, orthogonal on the real line, are well known. Quadrature formulas similar to the Gauss formulas for orthogonal polynomials have been studied. We generalize to the case of arbitrary complex poles and study orthogonality on a finite interval. The zeros of the orthogonal rational functions are shown to satisfy a quadratic eigenvalue problem. In the case of real poles, these zeros are used as nodes in the quadrature formulas

Restrictions on implicit filtering techniques for orthogonal projection methods

AbstractWe consider the class of the Orthogonal Projection Methods (OPM) to solve iteratively large eigenvalue problems. An OPM is a method that projects a large eigenvalue problem on a smaller subspace. In this subspace, an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method. Examples of OPMs are the Arnoldi and the Davidson method. We show how an OPM can be restarted — implicitly and explicitly. This restart can be used to remove a specific subset of vectors from the approximation subspace. This is called explicit filtering. An implicit restart can also be combined with an implicit filtering step, i.e. the application of a polynomial or rational function on the subspace, even if inaccurate arithmetic is assumed. However, the condition for the implicit application of a filter is that the rank of the residual matrix must be small

Quadrature formulas on the unit circle based on rational functions

AbstractQuadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szegő polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made

Convergence of modified approximants associated with orthogonal rational functions

AbstractLet {αn} be a sequence in the unit disk D = {z ∈ C: ¦z¦ < 1} consisting of a finite number of points cyclically repeated, and let L be the linear space generated by the functions Bn(z) = Πk=0n − αk(z − αk)¦αk¦(1 − αkz). Let {ϕn(z)} be orthogonal rational functions obtained from the sequence {Bn(z)} (orthogonalization with respect to a given functional on L), and let {ψn(z)} be the corresponding functions of the second kind (with superstar transforms ϕn∗(z) and ψn∗(z) respectively). Interpolation and convergence properties of the modified approximants Rn(z, un, vn) = (unψn(z) − vnψn∗(z))(unϕn(z) + vnϕn∗(z)) that satisfy ¦un¦ = ¦vn¦ are discussed

Explicit solutions for second order operator differential equations with two boundary value conditions

AbstractBoundary value problems for second order operator differential equations with two boundary value conditions are studied. Explicit expressions of the solutions in terms of data problems are given. By means of the application of algebraic techniques, analogous expressions to the ones known for the scalar case are obtained

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