Trigonometric orthogonal systems and quadrature formulae

Quadrature rules with maximal even trigonometric degree of exactness are considered. We give a brief historical survey on such quadrature rules. Special attention is given on an approach given by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (3) (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, pp. 31–54]. The main part of the topic is orthogonal trigonometric systems on [0,2π) (or on [−π,π)) with respect to some weight functions w(x). We prove that the so-called orthogonal trigonometric polynomials of semi-integer degree satisfy a five-term recurrence relation. In particular, we study some cases with symmetric weight functions. Also, we present a numerical method for constructing the corresponding quadratures of Gaussian type. Finally, we give some numerical examples. Also, we compare our method with other available methods.