Orthogonal Laurent polynomials and two-point Padé approximants associated with Dawson's integral

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

“…, m = 2n+1, respectively, is of paramount importance: if both these conditions are not satisfied, then the 'length' of the recurrence relations may be greater than three [16] (see, also, [24]). …”

“…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.…”

“…The main question regarding the convergence of two-point Padé approximants for this class of functions is with which rate it takes place, that is, the so-called quantitative result [42]: this necessitates obtaining results for the asymptotic behaviour (as n → ∞) of the orthonormal Lpolynomials φ n (z) in the entire complex plane. The theory of orthogonal L-polynomials is a natural framework for developing the theory of two-point Padé approximants, for both the scalar and matrix cases [24,[41][42][43][44]; (4) it turns out that, unlike the (finite) non-relativistic Toda lattice, whose direct and inverse spectral transform was constructed by Moser [45], and which is based on the theory of orthogonal polynomials and tri-diagonal Jacobi matrices, the direct and inverse scattering transform for the (finite) relativistic Toda lattice, introduced by Ruijsenaars [46], is based on the theory of orthogonal L-polynomials and pairs of bi-diagonal matrices [47] (see, also, [48]); and (5) for a finite, countable or uncountable index set K, let {ς p , p ∈ K} ⊂ C + := {z ∈ C; Im(z) > 0}, with ς p ς q ∀ p q ∈ K, and {̟ p , p ∈ K} ⊂ C be given point sets. A function F(z) which is analytic for z ∈ C + , with Im(F(z)) 0, is called a Nevanlinna function.…”

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

“…2m+1 > 0 for each m ∈ Z + 0 (see Equations (1.8) below, and Subsection 2.2, the proof of Lemma 2.2.2); and this fact is used, with little or no further reference, throughout this work (see, also, [24]).…”

“…respectively, is of paramount importance: if both these conditions are not satisfied, then the 'length' of the recurrence relations may be greater than three [16] (see, also, [24]).…”