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Asymptotics of $L^r$ extremal polynomials for ${0<r\leq\infty}$ on $C^{1+}$ Jordan regions (2502.17616v1)

Published 24 Feb 2025 in math.CA

Abstract: We study strong asymptotics of $Lr$-extremal polynomials for measures supported on Jordan regions with $C{1+}$ boundary for $0<r<\infty$. Using the results for $r=2$, we derive asymptotics of weighted Chebyshev and residual polynomials for upper-semicontinuous weights supported on a $C{1+}$ Jordan region corresponding to $r=\infty$. As an application, we show how strong asymptotics for extremal polynomials in the Ahlfors problem on a $C{1+}$ Jordan region can be obtained from that for the weighted residual polynomials. Based on the results we pose a conjecture for asymptotics of weighted Chebyshev and residual polynomials for a $C{1+}$ arc.

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