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1.
A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples. 相似文献
2.
《Journal of Computational and Applied Mathematics》2001,127(1-2):67-91
Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence. 相似文献
3.
Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed
under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid
for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on
a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules
as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature.
The work of the first author was supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under
grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM and Comunidad de Madrid under grant CCG06-UPM/MTM-539. The work of
the second author was partially supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under
grant MTM2005-08571. 相似文献
4.
Dr. Augustin Paulik 《Numerische Mathematik》1976,27(4):395-405
Summary In the present paper we discuss the optimal quadrature rules for integration with positive continuous weight function in Hardy space H2 of functions analytic in a circle of the complex plane. The new representations of the optimal weights and the norm of the error functional as functions of the nodes are obtained. On this basis we give an elementary proof for the existence of the optimal quadrature formula with free nodes. 相似文献
5.
In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described. 相似文献
6.
In this paper we obtained the asymptotic formula of the orthogonal rational function on the unit circle with respect to the
weight function μ(z) with preasigned poles, which are in the exterior of the unit disk. 相似文献
7.
Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational
interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding
interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct
two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the
Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields
an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger
than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical tests. 相似文献
8.
We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [–, ]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416. 相似文献
9.
Adhemar Bultheel 《Journal of Computational and Applied Mathematics》2010,235(4):927-949
We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur-Nevanlinna-Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur-Nevanlinna-Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna-Pick type for matricial Schur functions. 相似文献
10.
Wendell Ressler 《The Ramanujan Journal》2016,41(1-3):323-334
In this paper, we continue our work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper. 相似文献
11.
Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon’s work on right limits of power series, Baladi-Marmi-Sauzin recently introduced the notion of renascent right limit and rrl-continuation. We discuss a few examples and consider particularly the classical example of Poincaré simple pole series in this light. These functions are represented in the disc as series of infinitely many simple poles located on the circle; they appear, for instance, in small divisor problems in dynamics. We prove that any such function admits a unique rrl-continuation, which coincides with the function obtained outside the disc by summing the simple pole expansion. We also discuss the relation with monogenic regularity in the sense of Borel. 相似文献
12.
A. V. Abrahamyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):1-7
The paper is devoted to some properties of orthogonal on the unit circle rational functions with fixed poles. 相似文献
13.
A. V. Abramyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(5):253-257
The paper is devoted to investigation of the asymptotic behavior of orthogonal on the unit circle rational functions with
fixed poles. 相似文献
14.
In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the
real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory
type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications
related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong
Stieltjes weight functions. Several numerical experiments are finally carried out. 相似文献
15.
在本文中,我们首先给出一些基本的结果和一些概念,然后给出单位圆上带Cheby shev权的一些Cauchy主值积分的求积公式,最后给出了它们的误差估计. 相似文献
16.
Joris Van Deun Adhemar Bultheel Pablo Gonzá lez Vera. 《Mathematics of Computation》2006,75(253):307-326
We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside . Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order . This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on with arbitrary real poles outside this interval.
17.
《Journal of Computational and Applied Mathematics》1997,81(1):83-99
We consider interpolatory quadrature formulae, relative to the Legendre weight function on [−1, 1], having as nodes the zeros of the nth-degree Chebyshev polynomial of the third or fourth kind. Szegö has shown that the weights of these formulae are all positive. We derive explicit formulae for the weights, and subsequently use them to establish the convergence of the quadrature formulae for functions having a monotonic singularity at one or both endpoints of [−1, 1]. Moreover, we generate two new quadrature formulae, by adding 1, −1 to the sets of nodes considered previously, and show that these new formulae have almost all weights positive, exceptions occurring only among the weights corresponding to 1, −1. Also, we determine the precise degree of exactness of all the quadrature formulae in consideration, we obtain asymptotically optimal error bounds for these formulae, and show that almost all of them are nondefinite, exceptions occurring only among the formulae with a small number of nodes. 相似文献
18.
This paper is concerned with the numerical integration of functions with poles near the interval of integration. A method is given for modifying known quadrature rules, to obtain rules which are exact for certain classes of rational functions. 相似文献
19.
20.
A. V. Abramyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(2):67-78
The paper is devoted to the algebraic properties of rational functions which are orthogonal on the unit circle and have fixed poles. 相似文献