arXiv subjects

M. Yattselev

Publications and source records attributed to M. Yattselev.

3 recordsLinked to original sources

arXiv2024

n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

Let $ D $ be a bounded Jordan domain and $ A $ be its complement on the Riemann sphere. We investigate the $ n $-th root asymptotic behavior in $ D $ of best rational approximants, in the uniform norm on $ A $, to functions holomorphic on $ A $ having a multi-valued continuation to quasi every point of $ D $ with finitely many branches. More precisely, we study weak$^*$ convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $ n $-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.

math.CV↗

arXiv2008

Ratios of Norms for Polynomials and Connected n-width Problems

Let G be a bounded simply connected domain and E be a regular compact subset of G with connected complement. We investigate the asymptotic behavior of the Kolmogorov k-width, k=k(n), of the set of polynomials of degree at most n having the supremum norm at most 1 on G restricted to E in the space of continuous functions on E.

math.CA↗

arXiv2008

Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities

We study diagonal multipoint Padé approximants to sums of a Cauchy transform of a complex measure and a rational function. The measure is assumed to have compact regular support included into the real line and an argument of bounded variation on the support. For interpolation sets whose normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution, we show that the counting measures of poles of the approximants converge to the balayage of that distribution onto the support of the measure, in the weak-star sense, that the approximants themselves converge in capacity to the approximated function outside the support of the measure, and that the poles of the additional rational function attract at least as many poles of the approximants as their multiplicity and not much more.

math.CA↗