On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

Abstract

Abstract. We consider the convergence of Gauss-type quadrature formulas for the integral ∞ 0 f (x)ω(x)dx, where ω is a weight function on the half line [0, ∞). The n-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomialsIt is proved that under certain Carleman-type conditions for the weight and when p(n) or q(n) goes to ∞, then convergence holds for all functions f for which fω is integrable on [0, ∞). Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

“…On the other hand, as pointed out in Section 1, quadrature formulas exactly integrating Laurent polynomials appeared as a working tool related to the solution of the so-called strong moment problems. During the last years, some of the present authors have considered such quadratures from the optic of a numerical integration approach, carrying out a series of numerical experiments ([8], [2], [3]) and emphasizing their intimate relation to the theory of orthogonal Laurent polynomials and two-point Padé approximants. In this paper, we have intended to follow this line so that some results as, e.g., Proposition 4.5 which could have been deduced from the works by Ranga and collaborators (see, e.g., [30]), have now been revisited starting from the theory of orthogonal Laurent polynomials.…”

“…To conclude this section, it should be remarked that the effectiveness and numerical power of the quadrature rules (2.7) in order to estimate integrals like (2.6) have been displayed recently in [2,8]. On the other hand, from the above theorems one sees that the basis to compute L-orthogonal formulas is the knowledge of the sequence {ψ n } n≥0 .…”

Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

“…This kind of quadratures can be considered as a particular case of those studied by some of the present authors in [6].…”

“…Further results on convergence of the L-orthogonal formula can be deduced from [6] and [7]. Concerning convergence of 2PA's see [16,27].…”

“…So, this is just the starting point of the present paper: to develop a general theory on some aspects of orthogonal Laurent polynomials, with special emphasis on quadrature and Padé approximation, so that known results on either orthogonal polynomials or certain sequences of orthogonal Laurent polynomials already studied can now be deduced in a straightforward way. For an alternative approach on this topic, via orthogonal polynomials with respect to varying distributions, see [27] and also the series of papers [3,4,5,6] by some of the present authors. On the other hand, related results can be found in [31] and [32] in connection with certain classes of continued fractions.…”