On weight functions which admit explicit Gauss-Turán quadrature formulas

Abstract

Abstract. The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding s-orthogonal polynomials, of the same degree, are independent of s. These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.

Error bounds for Gauss-Turán quadrature formulae of analytic functions

“…On the basis of (3.10) we conclude that the corresponding Gauss-Turán quadrature formulae converge if s is a fixed integer and n → +∞, since [16] have introduced for each n a class of weight functions defined on [−1, 1] for which explicit Gauss-Turán quadrature formulas can be found for all s. Indeed, these classes of weight functions have the peculiarity that the corresponding s-orthogonal polynomials, of the same degree, are independent of s. This class includes certain generalized Jacobi weight functions…”

Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument

“…Hammer and Wicke [57,58,59,60] are not advantageous in our case. A cubic spline interpolation f (x) = 3 n=0 c n x n induces the higher moments…”

Monotonicity of the error term in Gauss-Turán quadratures for analytic function

“…However, such weights depend on s. The weight function in (d) can be omitted from investigation because W n (-t) = (-1)" V n (r). Gori and Micchelli [6] have introduced for each h a class of weight functions defined on [-1, 1] for which the explicit Gauss-Turan quadrature formulae can be found for all s. In other words, these classes of weight functions have the peculiarity that the corresponding ^-orthogonal polynomials, of the same degree, are independent of s. This class includes certain generalized Jacobi weight functions where U n -\(cos9) = sinnO/ sin9 (a Chebyshev polynomial of the second kind) and fx > -1. In this case, the Chebyshev polynomials T n {t) appear to be s-orthogonal polynomials.…”