arXiv subjects
Publications and source records attributed to Sylvain Chevillard.
We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.
Performing a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. When the unstable part lies significantly above the interpolation error of the method, the circuit is considered unstable. Working with a projection provides one, at small cost, with a first appraisal of the unstable part of the system. Both small-signal and large-signal stability analysis can be performed with this projection-based method. In the small-signal case, a low-order rational approximation can be fitted on the unstable part to find the location of the unstable poles.
We derive some lower bounds in rational approximation of given degree to functions in the Hardy space $H^2$ of the disk. We apply these to asymptotic errors rates in approximation to Blaschke products and to Cauchy integrals on geodesic arcs. We also explain how to compute such bounds, either using Adamjan-Arov-Krein theory or linearized errors, and we present a couple of numerical experiments on several types of functions. We dwell on the Adamjan-Arov-Krein theory and a maximin principle developed in the article "An L^p analog of AAK theory for p \textgreater{}= 2", by L. Baratchart and F. Seyfert, in the Journal of Functional Analysis, 191 (1), pp. 52-122, 2012.
The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x > 0 due to cancellation. Based on a method recently proposed by Gawronski, Müller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor coefficients of G turn out to obey an ill-conditioned three-term recurrence. We use the classical Miller algorithm to overcome this issue. We bound all errors and our implementation allows an arbitrary and certified accuracy, that can be used, e.g., for providing correct rounding in arbitrary precision.