Orthogonal rational functions and rational modifications of a measure on the unit circle

AbstractIn this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure

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

Down on the Brandywine

https://digitalcommons.library.umaine.edu/mmb-vp/1352/thumbnail.jp

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

An atypical presentation of myositis ossificans

In the following case study an atypical presentation of myositisossificans (MO) in the superior anterolateral thigh of a youngsoccer player is discussed. This case demonstrates that MO canpresent without obvious history of trauma, which makes thediagnosis of this condition more challenging. The most importantdifferential diagnosis is malignant osteosarcoma or soft-tissuesarcoma, which usually presents without trauma. Additionallyboth pathologies typically occur in the same population.Keywords: case report, ossification, osteosarcom

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

That Aeroplane Glide

https://digitalcommons.library.umaine.edu/mmb-vp/6659/thumbnail.jp