Rational Gauss-Chebyshev quadrature formulas for complex poles outside $[-1,1]$

Abstract

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [−1, 1] to arbitrary complex poles outside [−1, 1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1].

The existence and construction of rational Gauss-type quadrature rules

“…⊂ D, so that β k = J inv (α k ). The so-called Chebyshev nORFs (with respect to the Chebyshev weight function w(x) and inner product f , g w = J w (f g c )), with arbitrary complex poles outside I, are then given by (see [8])…”

“…[1,Chapt. 11.6] and [2,7,8,10,11,12,14,15,17,18,19,20]. Here we can also consider more general rational Gauss-type quadrature rules obtained by fixing one or two nodes in the quadrature rule.…”

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

“…In [17,Lem. 3.1] a relation has been proven between the factors Z k (x) and the Blaschke factors ζ k * (z) and 1/ζ k (z), so that for…”

“…Based on this relation and the property of orthonormality given by (1), the following explicit expressions for the Chebyshev ORFs ϕ n (x) related to the i th weight in Table 1 have been proven in [17,Thm. 3.2].…”

“…Further, in [17] the expressions of the nodes and weights for the Gauss-Chebyshev quadrature formulas, as well as the theorem on the asymptotic distribution of the nodes, are extended to the case of complex poles, arbitrary but fixed outside [−1, 1]. The main purpose of this paper is to extend the algorithm in [16] to this case.…”