2026/07/23 by Sampad Lahiry
Mathematics · #math.CA #math.CV
We study the Chebyshev extremal problem on a compact Riemann surface X of genus g>0. As an analog to monic polynomials, we consider admissible meromorphic functions having no poles away from a marked point P∞ (with adequate normalisation). For a nonpolar compact set E⊂ X∖\P∞\, we show that the n-th root of the Chebyshev constant tn(E) converges to the capacity of E, and we establish the corresponding Bernstein-Walsh inequality. In the second part of the paper, using a Cauchy kernel adapted to Riemann surfaces, we study the refined Szegő--Widom asymptotics for the extremals as well as the Widom factors Wn(E)=(tn(E))/(cap(E)n), assuming that E is a finite union of p closed discs with analytic boundaries. The geometry of the Schottky double, which has genus 2g+p-1, enters explicitly into these asymptotics. We find that there is no analogue of Faber-type asymptotics even for one single boundary curve. We conclude by constructing explicit examples in genus 1 using the Weierstrass-\wp function.