Extremal Polynomials with Preassigned Zeros and Rational Approximants

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

“…We have considered the particular case w ≡ 1, α = 1, β = 2. Numerical experiments show that the zeros of the Stieltjes polynomials for this measure have an interesting behaviour; while the zeros of S 2n sit on [−2, −1] ∪ [1,2], those of S 2n+1 draw the level curve {ζ ∈ C : g Ω (ζ, ∞) = g Ω (0, ∞)}. Figure 1 shows the zeros of S n for n = 20, 21 (the small circles are the zeros of S 20 and the crosses the zeros of S 21 ).…”

“…[2] and [3]). This is done in order to ensure that the polynomials, whose zeros are the free poles of the rational approximant, be orthogonal with respect to a positive measure.…”

“…Since S n is orthogonal with respect to a sign changing function, equations (2), in general, do not guarantee that the zeros of S n lie in [−1, 1], that they are simple and distinct from the zeros of p n−1 , or even that they are real. However, for the ultraspherical weight function w λ , w λ (x) = (1 − x 2 ) λ−1/2 , 0 ≤ λ ≤ 2, Szegő proved in [24] that these properties hold for all n. Positivity of the coefficients appearing in the quadrature formula and interlacing properties of the zeros have also been studied for the ultraspherical weights w λ , 0 ≤ λ ≤ 1, in [11] and [4], respectively.…”

Multipoint Padé-Type Approximants. Exact Rate of Convergence

“…A key question in the solution of the problems above is the study of equilibrium distributions in the presence of exterior fields Our paper is particularly influence by the results contained in [4] which are ready to use for our purpose. We also refer the reader to the excellent monograph [8] whose methods may also be applied to obtain similar results to the ones we give here under weaker assumptions (this is the main strategy of [1] in contrast with [2]). The results in [8 (see Section 3.3)] must be adapted in order that they cover the type of varying weights we consider here.…”

“…Later, in the same framework, we studied the problem when lim n m(n)/n = θ ∈ [0, 1] (see [2]). Recently, we received a preprint of A. Ambroladze and H. Wallin [1] where they consider, also in the classical setting, the problem of upper estimates for…”