Two-point Padé expansions for a family of analytic functions

Jones, William B.; Njåstad, Olav; Thron, W. J.

doi:10.1016/0377-0427(83)90034-1

Cited by 76 publications

(44 citation statements)

References 17 publications

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“…Jones, O. Njåstad and W. Thron [12,11], L. Cochran and S. Cooper [6], S. Ranga [19], and G. López-Lagomasino [15]. Very deep investigations of the special case f (z) = 1/(z − x) with z ∈ (0, ∞), a parameter, were given in [15] as a compilation of several papers [16,17,18] by López-Lagomasino.…”

Section: Gauss-type Quadrature 723mentioning

confidence: 99%

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

et al. 1999

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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.

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Section: Gauss-type Quadrature 723mentioning

confidence: 99%

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

et al. 1999

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“…Properties of doubly infinite moment sequences and corresponding quadrature formulas and continued fractions expansions are also treated in [6].…”

Section: J Imentioning

confidence: 99%

“…The Pade approximants at oo for the series E£L O C « Z~" a r e closely related to the same orthogonal polynomials. The integral jjd\l/(t)/{t -z) defines an analytic 6 Olav Njastad and W. J. Thron [2 ] function F(z) outside /, and the series E^= o c n z~" is an asymptotic expansion for F{z) at oo with respect to suitable regions. The sequence of main diagonal Pade approximants has subsequences converging to functions of the form G(z) = zj r d<j>(t)/(t -z), where the functions <> are among the distribution functions giving rise to the same moment sequence { c n : n = 0,1,2,...} as >//.…”

Section: Introductionmentioning

confidence: 99%

Unique solvability of the strong Hamburger moment problem

Njåstad

Thron

1986

J Aust Math Soc A

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Methods from the theory of orthogonal polynomials are extended to L-polynomials Y. q n = p a n z". By this means the authors and W. B. Jones (J. Math. Anal. Appl. 98 (1984), 528-554) solved the strong Hamburger moment problem, that is, given a double sequence {c n }°? x , to find a distribution function i|/(r), non-decreasing, with an infinite number of points of increase and bounded on -oo < t < oo, such that for all integers n, c n = f™ x (-t) n dip(t). In this article further methods such as analogues of the Liouville-Ostrogradski formula and of the Christoffel-Darboux formula are developed to investigate when the moment problem has a unique solution. This will be the case if and only if a sequence of nested disks associated with the sequence { c n }™ x has only a point as its intersection (the so called limit point case).

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“…Their study, not only suffered a rapid development in the last decades giving rise to a theory of orthogonal Laurent polynomials on the real line (see e.g. [8], [14], [24], [28], [33] and [36]), but it was extended to an ampler context leading to a general theory of orthogonal rational functions (see [2]). …”

Section: Introductionmentioning

confidence: 99%

A matrix approach to the computation of quadrature formulas on the unit circle

Cantero¹,

Cruz-Barroso²,

González-Vera³

2008

Applied Numerical Mathematics

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In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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Two-point Padé expansions for a family of analytic functions

Cited by 76 publications

References 17 publications

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

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

Unique solvability of the strong Hamburger moment problem

A matrix approach to the computation of quadrature formulas on the unit circle

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