In this paper, we investigate a numerical method for the construction of an optimal set of quadrature rules in the sense of Borges (Numer. Math. 67, 271–288, 1994) for two or three definite integrals with the same integrand and interval of integration, but with different weight functions, related to an arbitrary multi-index. The presented method is illustrated by numerical examples.
In this paper an error estimate for quadrature rules with an even maximal trigonometric degree of exactness (with an odd number of nodes) for $$2\pi -$$ periodic integrand, analytic in a circular domain, is given. Theoretical estimate is illustrated by numerical example.
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 [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...
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