Banach Journal of Mathematical Analysis

A special Gaussian rule for trigonometric polynomials

Aleksandar S. Cvetkovic, Gradimir V. Milovanovic, and Marija P. Stanic

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Abram Haimovich Turetzkii [Uchenye Zapiski, 1 (149) (1959), 31-55 (translation in English in East J. Approx. 11 (2005), 337-359)] considered interpolatory quadrature rules which have the following form $\int_0^{2\pi}f(x)w(x)d x\approx \sum_{\nu=0}^{2n}w_\nu f(x_\nu)$, and which are exact for all trigonometric polynomials of degree less than or equal to $n$. Maximal trigonometric degree of exactness of such quadratures is $2n$, and such kind of quadratures are known as quadratures of Gaussian type or Gaussian quadratures for trigonometric polynomials. In this paper we prove some interesting properties of a special Gaussian quadrature with respect to the weight function $w_m(x)=1+\sin mx$, where $m$ is a positive integer.

Article information

Banach J. Math. Anal., Volume 1, Number 1 (2007), 85-90.

First available in Project Euclid: 21 April 2009

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Zentralblatt MATH identifier

Primary: 65D30: Numerical integration
Secondary: 42A05: Trigonometric polynomials, inequalities, extremal problems

trigonometric polynomials of semi-integer degree orthogonality Gaussian type quadratures


Milovanovic, Gradimir V.; Cvetkovic, Aleksandar S.; Stanic, Marija P. A special Gaussian rule for trigonometric polynomials. Banach J. Math. Anal. 1 (2007), no. 1, 85--90. doi:10.15352/bjma/1240321558.

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  • A.S. Cvetković and G.V. Milovanović, The Mathematica Package “OrthogonalPolynomials”, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36.
  • G.V. Milovanović, A.S. Cvetković and M.P. Stanić, Trigonometric orthogonal systems and quadrature formulae, preprint.
  • G.V. Milovanović, A.S. Cvetković and M.P. Stanić, Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree, Appl. Math. Comput. (to appear).
  • G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore–New Jersey–London–Hong Kong, 1994.
  • 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).