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Quadrature rules of Gaussian type for trigonometric polynomials with preassigned nodes
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An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges' sense [Numer. Math. 67 (1994), 271-288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonometric multiple orthogonal polynomials of semiinteger degree. Finally, theoretical results are illustrated by some numerical examples.
In this paper we consider multiple orthogonal polynomials on the real line, defined using orthogonality conditions spread out over r different measures. Such polynomials are generalization of the ordinary orthogonal polynomials (case r = 1).
Kronrod in 1964, trying to estimate economically the error of the n-point Gauss quadrature formula for the Legendre weight function, developed a new formula by adding to the n Gauss nodes n + 1 new ones, which are determined, together with all weights, such that the new formula has maximum degree of exactness. It turns out that the new nodes are zeros of a polynomial orthogonal with respect to a variable-sign weight function, considered by Stieltjes in 1894, without though making any reference to quadrature. We survey the considerable research work that has been emerged on this subject, during the past fifty years, after Kronrod's original idea.
In this paper we consider multiple orthogonal trigonometric polynomials of semi-integer degree, which are necessary for constructing of an optimal set of quadrature rules with an odd number of nodes for trigonometric polynomials in Borges’ sense [Numer. Math. 67 (1994) 271-288]. We prove that such multiple orthogonal trigonometric polynomials satisfy certain recurrence relations and present numerical method for their construction, as well as for construction of mentioned optimal set of quadrature rules. Theoretical results are illustrated by some numerical examples.
In this paper, we give a brief survey of orthogonal trigonometric polynomials of semi-integer degree with respect to some weight functions w on [ − π, π). Such orthogonal systems are connected with interpolatory quadrature rules with an even maximal trigonometric degree of exactness.
In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on (-π,π) for integrand analytic in a certain domain of complex plane. Copyright ©2013 John Wiley & Sons, Ltd.
The set of all quadrature formulae for computing an integral in the interval [0,2π] with non-negative weight, having a trigonometric degree of accuracy n in the case of n + 1 nodes, is described.
Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x) on [0,2π) have been considered firstly by Turetzkii in [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (2005) 337–359 (Translation in English from Uchenye Zapiski, Vypusk 1(149), Seria math. Theory of Functions, Collection of Papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk (1959) 31–54)]. It is proved that such orthogonal trigonometric polynomials of semi-integer degree satisfy five-term recurrence relation. In this paper we present explicit formulas for five-term recurrence coefficients for some weight functions.
Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x)w(x) on [0,2π)[0,2π) have been considered firstly by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (2005) 337–359 (translation in English from Uchenye Zapiski, Vypusk 1(149), Seria Math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, (1959) pp. 31–54)]. Such orthogonal systems are connected with quadrature rules with an even maximal trigonometric degree of exactness (with an odd number of nodes), which have application in numerical integration of 2π2π-periodic functions. In this paper we study asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree with respect to a strictly positive weight function satisfying the Lipschitz-Dini condition.
In this paper we consider trigonometric polynomials of semi-integer degree orthogonal with respect to a linear functional, defined by a nonnegative Borel measure. By using a suitable vector form we consider the corresponding Fourier sums and reproducing kernels for trigonometric polynomials of semiinteger degree. Also, we consider the Christoffel function, and prove that it satisfies extremal property analogous with the algebraic case.
We introduce orthonormal trigonometric polynomials of semiinteger degree with respect to a weight function on [-π,π) and prove the Christoffel-Darboux formula for such an orthonormal trigonometric system.
In this paper, the algebraic construction of quadrature formulas for weigh-ted periodic integrals is revised. For this purpose, two classical papers ([10] and [14]) in the literature are revisited and certain relations and connections are brought to light. In this respect, the concepts of "bi-orthogonality" and "para-orthogonality" are shown to play a fundamental role.
In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up
to degree as high as possible are constructed in order to approximate 2π-periodic weighted integrals. For this purpose, certain bi-orthogonal systems of trigonometric functions are introduced and
their most relevant properties studied. Some illustrative numerical examples are also given. The paper completes the results
previously given by Szegő in Magy Tud Akad Mat Kut Intez Közl 8:255–273, 1963 and by some of the authors in Annales Mathematicae et Informaticae 32:5–44, 2005.
Quadrature rules with maximal even trigonometric degree of exactness are considered. We give a brief historical survey on such quadrature rules. Special attention is given on an approach given by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (3) (2005) 337–359. Translation in English from Uchenye Zapiski, Vypusk 1 (149). Seria Math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, 1959, pp. 31–54]. The main part of the topic is orthogonal trigonometric systems on [0,2π) (or on [−π,π)) with respect to some weight functions w(x). We prove that the so-called orthogonal trigonometric polynomials of semi-integer degree satisfy a five-term recurrence relation. In particular, we study some cases with symmetric weight functions. Also, we present a numerical method for constructing the corresponding quadratures of Gaussian type. Finally, we give some numerical examples. Also, we compare our method with other available methods.
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal , . A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented. Comment: Submitted by F. Peherstorfer to Math. Comp
. We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to find a finite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene. These values are then used to determine the RGB coordinates of the light by evaluating three definite integrals, each with a common integrand (the SPD) and interval of integration but with distinct weight functions. We develop a method for selecting the sample wavelengths in an optimal manner. More abstractly, we examine the problem of numerically evaluating a set of m definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration. It is shown that when m 3 it is not efficient to use a set of m Gauss rules because valuable information is wast...
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