Trigonometric orthogonal systems and quadrature formulae

  • Authors:

  • Gradimir V. Milovanović;Aleksandar S. Cvetković;Marija P. Stanić

  • Affiliations:

  • Department of Mathematics, Faculty of Electronic Engineering, University of Niš, P.O. Box 73, 18000 Niš, Serbia;Department of Mathematics, Faculty of Electronic Engineering, University of Niš, P.O. Box 73, 18000 Niš, Serbia;Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2008

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Abstract

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 [0,2@p) (or on [-@p,@p)) 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.