Asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree

Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x)w(x) on [0,2π)[0,2π) have been considered firstly by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (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)]. Such orthogonal systems are connected with quadrature rules with an even maximal trigonometric degree of exactness (with an odd number of nodes), which have application in numerical integration of 2π2π-periodic functions. In this paper we study asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree with respect to a strictly positive weight function satisfying the Lipschitz-Dini condition.