Trigonometric orthogonal systems and quadrature formulae

Abstract

a b s t r a c tQuadrature 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.

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

“…It is known that such a trigonometric polynomial of semi-integer degree has 2n+1 distinct simple zeros in [−π, π) (see [8]). So, y * ∈ S 2n and y * = T n (x).…”

“…2n be zeros of orthogonal trigonometric polynomial of semi-integer degree n + 1/2 (for more details see [8]). Then (x (0)…”

“…We choose the weight functions w(x) = 1 + sin mx, m ∈ N, for our numerical experiments because we have explicit formulas for five-term recurrence coefficients for the corresponding orthogonal trigonometric polynomials of semi-integer degree, which are necessary for constructing Gaussian type quadrature rules (3.2) (see [8,9]). …”

“…3]) and it has in [−π, π) exactly 2n + 1 distinct simple zeros. Numerical methods for constructing orthogonal trigonometric polynomials of semi-integer degree and the corresponding quadrature formula (1.2) have been given in [8].…”

Generalisane kvadraturne formule Gauss-ovog tipa

“…stepena k +1/2, k ∈ Z, na intervalu [0, 2π) u odnosu na neku težinsku funkciju w. Neke osnovne osobine trigonometrijskih polinoma polu-celobrojnog stepena date su u [91]. Oni su detaljno analizirani u [59] i [60], gde je dat i numerički metod za njihovu konstrukciju. Dobijeni rezultati su dalje generalisani na s− i σ−ortogonalne trigonometrijske polinome polu-celobrojnog stepena.…”