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1.
Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure
In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials. 相似文献
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We prove a necessary and sufficient condition for integrability of the reciprocal weight function of orthogonal polynomials. The condition is given in terms of the asymptotic behaviour of the norm of extremal polynomials with prescribed coefficients. 相似文献
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A new numerical quadrature formula on the unit circle 总被引:1,自引:0,他引:1
In this paper we study a quadrature formula for Bernstein–Szegő measures on the unit circle with a fixed number of nodes and
unlimited exactness. Taking into account that the Bernstein–Szegő measures are very suitable for approximating an important
class of measures we also present a quadrature formula for this type of measures such that the error can be controlled with
a well-bounded formula.
This work was supported by Ministerio de Educación y Ciencia under grants number MTM2005-01320 (E. B. and A. C.) and MTM2006-13000-C03-02
(F. M.). 相似文献
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E. Berriochoa Esnaola A. Cachafeiro López J.R. Illán-González E. Martínez-Brey 《Applied mathematics and computation》2011,218(8):4437-4447
In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula. 相似文献
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A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle. 相似文献
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We characterize the supports of the measures having quadrature formulae with similar exactness as Gauss’ theorem. Indeed we obtain the supports of the measures from which an m-point quadrature formula can be obtained such that it exactly integrates functions in the space ? m?k,m?k [ $ \bar z $ , z]. We also give a method for obtaining the nodes and the quadrature coefficients in all the cases and, as a consequence, we solve the same problem in the space of trigonometric polynomials. 相似文献
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Strong (or Szeg
-type) asymptotics for orthogonal polynomials with respect to a Sobolev inner product with general measures (the first measure is arbitrary and the second one is absolutely continuous and satisfying a smoothness condition) is obtained. Examples, illustrating the theorems proved, are presented. 相似文献
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Elías Berriochoa Alicia Cachafeiro José M. García-Amor 《Complex Analysis and Operator Theory》2012,6(3):651-664
We study the Hermite interpolation problem with equally spaced nodes on the unit circle. We obtain new conditions for the derivatives in order that the Hermite interpolants uniformly converge to continuous functions. As a consequence we obtain some improvements in the case of the bounded interval. 相似文献