Szegő quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane. However, often it is difficult to determine the quadrature error. Recently, Spalević introduced generalized averaged Gauss quadrature rules for estimating the quadrature error obtained when applying Gauss quadrature over an interval on the real axis. We describe analogous quadrat...

We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature form...

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that...

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory orthogonal polynomials its nodes weights can be computed efficiently by using tools numerical linear algebra. An interestin...