Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series
نویسندگان
چکیده
منابع مشابه
Trigonometric and Gaussian Quadrature
Some relationships are established between trigonometric quadrature and various classical quadrature formulas. In particular Gauss-Legendre quadrature is shown to be a limiting case of trigonometric quadrature. In an earlier paper [1] it was noted that there exist trigonometric and exponential analogs of Gaussian quadrature formulas. We now extend those results to show several interesting featu...
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An anti-Gaussian quadrature formula is an (n+ 1)-point formula of degree 2n− 1 which integrates polynomials of degree up to 2n+ 1 with an error equal in magnitude but of opposite sign to that of the n-point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show tha...
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Most nonclassical Gaussian quadrature rules are difficult to construct because of the loss of significant digits during the generation of the associated orthogonal polynomials. But, in some particular cases, it is possible to develop stable algorithms. This is true for at least two well-known integrals, namely ¡l-(Loêx)-x°f(x)dx and ¡Ô Em(x)f(x)-dx. A new approach is presented, which makes use ...
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We give several descriptions of positive quadrature formulas which are exact for trigonometric-, respectively, Laurent polynomials of degree less or equal to n − 1 − m, 0 ≤ m ≤ n − 1. 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 ...
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We show how to combine incidence matrices, which admit Hermite-Birkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the co...
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ژورنال
عنوان ژورنال: Constructive Mathematical Analysis
سال: 2019
ISSN: 2651-2939
DOI: 10.33205/cma.613948