Asymptotic Properties of Stieltjes Polynomials and Gauss-Kronrod Quadrature Formulas

AbstractStieltjes polynomials are orthogonal polynomials with respect to the sign changing weight function wPn(·, w), where wPn(·, w) is the nth orthogonal polynomial with respect to w. Zeros of Stieltjes polynomials are nodes of Gauss-Kronrod quadrature formulae, which are basic for the most frequently used quadrature routines with combined practical error estimate. For the ultraspherical weight function wλ(x) = (1 − x2)λ − 1/2, 0 ≤ λ ≤ 1, we prove asymptotic representations of the Stieltjes polynomials and of their first derivative, which hold uniformly for x = cos θ, ϵ ≤ π − ϵ, where ϵ ∈ (0, π/2) is fixed. Some conclusions are made with respect to the distribution of the zeros of Stieltjes polynomials, proving an open problem of Monegato [15, p. 235] and Peherstorfer [23, p. 186]. As a further application, we prove an asymptotic representation of the weights of Gauss-Kronrod quadrature formulae with respect to wλ, 0 ≤ λ ≤ 1, and we prove the precise asymptotical value for the variance of Gauss-Kronrod quadrature formulae in these cases

Can reindeer husbandry management slow down the shrubification of the Arctic?

Rapid climate change is threatening the stability and functioning of Arctic ecosystems. As the Arctic warms, shrubs have been widely observed to expand, which has potentially serious consequences for global climate regulation and for the ecological processes characterising these ecosystems. However, it is currently unclear why this shrubification has been spatially uneven across the Arctic, with herbivory being suggested as a key regulating factor. By taking advantage of freely available satellite imagery spanning three decades, we mapped changes in shrub cover in the Yamal Peninsula and related these to changes in summer temperature and reindeer population size. We found no evidence that shrubs had expanded in the study site, despite increasing summer temperatures. At the same time, herbivore pressure increased significantly, with the local reindeer population size growing by about 75%. Altogether, our results thus point towards increases in large herbivore pressure having compensated for the warming of the Peninsula, halting the shrubification of the area. This suggests that strategic semi-domesticated reindeer husbandry, which is a common practice across the Eurasian Arctic, could represent an efficient environmental management strategy for maintaining open tundra landscapes in the face of rapid climate change