Ultraspherical Gauss--Kronrod Quadrature Is Not Possible for $\lambda > 3$

Abstract

With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter λ has only few real zeros for λ > 3 and sufficiently large n. Since the nodes of the GaussKronrod quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that Gauss-Kronrod quadrature is not possible for λ > 3. On the other hand, for λ = 3 and sufficiently large n, even partially positive Gauss-Kronrod quadrature is possible.

Asymptotics for Stieltjes polynomials, padé-type approximants, and Gauss-Kronrod quadrature

“…Using Cauchy's integral formula he obtains an integral representation of the polynomial E n+1 (different from (18) and (19)) which allows him to prove that it satisfies full orthogonality relations with respect to p n (x) dx. Therefore, E n+1 is S n+1 up to a multiplicative constant.…”

“…[14] and [18]) that, in general, Stieltjes polynomials may have complex zeros. Despite this fact, statement a) of Theorem 1 shows that the zeros may only accumulate on E (on S(µ) if S(µ) = ess[b − a, b + a]) when µ ∈ Reg.…”

Stieltjes-type polynomials on the unit circle

“…In order to prove (23), let us show that the function s n,m (z) − 1/g n (z) z m+1 is analytic inside the unit circle. In fact, since g n (z) = 0 whenever |z| = 1, it can only have a singularity at z = 0 which is not the case since, according to (14) with L(z) = z −m , we know that (see also (21))…”

“…The ultraspherical measures dµ(x) = (1 − x 2 ) λ−1/2 dx, λ > −1/2, suffice to display the rich behavior of the zeros of E n+1 . Depending on λ, they may have exemplary zeros which are real, simple, and interlace the zeros of the corresponding orthogonal polynomial, or erratic when most of the zeros are complex (see [27,21,5]). …”

Generalized Stieltjes Polynomials and Rational Gauss?Kronrod Quadrature

“…[4], [7], and [1]; and the references therein). However, these properties are not satisfied in general (see, for instance, [8]). Until recently, little was known on the asymptotic behaviour of the Stieltjes polynomials outside the support of the measure.…”