Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part III. The unbounded case

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

“…This expression appears just by rewriting Theorem 2.3 in [4] in terms of orthogonal Laurent polynomials. Indeed, let ψ n be the n-th orthogonal L-polynomial normalized as in Section 2 and let f be a function Riemann-Stieltjes integrable with respect to φ such that f (2n) (t) is continuous for a ≤ t ≤ b.…”

“…Proceed as in [4,Theorem 5.3] to obtain the conclusion for a continuous functions f in [0, ∞) such that lim x→∞ f (x) exists, and after that, proceed as in [21,Lemma 2.7].…”

“…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.…”

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

“…It is important to note [14] that the classical and strong moment problems (SMP, HMP, SSMP, and SHMP) are special cases of a more general theory, where moments corresponding to an arbitrary, countable sequence of (fixed) points are involved (in the classical and strong moment cases, respectively, the points are ∞ repeated and 0, ∞ cyclically repeated), and where orthogonal rational functions [26,32,33] play the rôle of orthogonal polynomials and orthogonal Laurent (or L-) polynomials; furthermore, since L-polynomials are rational functions with (fixed) poles at the origin and at the point at infinity, the step towards a more general theory where poles are at arbitrary, but fixed, positions/locations in C ∪ {∞} is natural, with applications to, say, multi-point Padé, and Padé-type, approximants [24,[34][35][36][37][38]; the so-called Christoffel numbers [35]. When considering the computation of integrals of the form π −π g(e iθ ) dµ(θ), where g is a complex-valued function on the unit circle D := {z ∈ C; |z|= 1} and µ is, say, a positive measure on [−π, π], in particular, when g is continuous on D, keeping in mind that a function continuous on D can be uniformly approximated by Lpolynomials, it is natural to consider, instead of orthogonal polynomials, Laurent polynomials, which are also related to the associated trigonometric moment problem [35,39] (see, also, [40] where c l = R s l e −N V(s) ds, l ∈ Z, with respect to the (unbounded) domain {z ∈ C; ε | Arg(z)| π−ε}, where Arg( * ) denotes the principal argument of * , and ε > 0 is sufficiently small.…”

Orthogonal rational functions and quadrature on the real half line

“…In [3] (see also [10]) the polynomials were replaced by Laurent polynomials having a finite number of positive and negative powers of z. In that situation, exactness can be obtained for certain spaces of Laurent polynomials of dimension 2n − 1, if we take as nodes the zeros of orthogonal Laurent polynomials.…”