A special Gaussian rule for trigonometric polynomials

Abstract

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.