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) 337359. Translation in English from Uchenye Zapiski, Vypusk 1 (149). Seria Math. Theory of Functions, Collection of papers, Izdatelstvo Belgosuniversiteta imeni V.I. Lenina, Minsk, 1959, pp. 3154]. The main part of the topic is orthogonal trigonometric systems on [0,2?) [ 0 , 2 ? ) (or on [??,?) [ ? ? , ? ) ) with respect to some weight functions w(x) 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.
orthogonality trigonometric degree of exactness weights recurrence relation trigonometric interpolation quadrature rules of gaussian type nodes