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1.
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
2.
Adhemar Bultheel Pablo González-Vera Erik Hendriksen Olav Njåstad 《Numerical Algorithms》1992,3(1):91-104
Leta 1,...,a p be distinct points in the finite complex plane ?, such that |a j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ0, μ π (j) , ν π (j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?* outside {a 1,...,a p,b 1,...,b p}. 相似文献
3.
Ruymán Cruz-Barroso Leyla Daruis Pablo González-Vera Olav NjÅstad 《Journal of Computational and Applied Mathematics》2007
In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied. 相似文献
4.
In this paper quadrature rules introduced by Jagerman [1] and Stetter [2] are considered and asymptotic expansions for the error given. This allows to make use of the Romberg extrapolation process. Such rules can be viewed as generalizations of the well-known mid-point rule. Thus, numerical examples comparing these rules are finally presented. 相似文献
5.
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
6.
Adhemar Bultheel Pablo González-Vera Erik Hendriksen Olav Njåstad 《Numerical Algorithms》1992,2(1):85-114
We shall consider nested spacesl
n
,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,l
n
=
n
, the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research 相似文献
7.
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
8.
Leyla Daruis Pablo Gonzlez-Vera Francisco Marcelln 《Journal of Computational and Applied Mathematics》2002,140(1-2)
Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form
For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples. 相似文献
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9.
Juan A. González-Vera 《Tetrahedron》2007,63(37):9229-9234
New indole alkaloid analogues, containing a 10b-methyl- or a 10b-hydroxy-1,2,4,5,10b,10c-hexahydropyrrolo[1′,2′,3′:1,9a,9]imidazo[1,2-a]indole skeleton, have been obtained by highly stereoselective electrophile addition-cyclization reactions of a tryptophan-derived α-amino nitriles. 相似文献
10.