arXiv subjects
Publications and source records attributed to Benedikt Buchecker.
We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.
The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $μ$ supported on a Jordan arc $Γ$ is \[ λ_n(μ,z_0)=\inf\left\{\int_Γ|P|^2dμ\mid P\text{ is a polynomial of degree at most }n\text{ and } P(z_0)=1\right\}. \] It is natural to extend this notion to $z_0=\infty$ and define $λ_n(μ,\infty)$ to be the infimum of the squared $L^2(μ)$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of $λ_n(μ,z_0)$ for $|z_0|>1$ and $z_0=\infty$ and arbitrary finite measures supported on the unit circle. Widom has proved a version of Szegő's theorem for measures supported on $C^{2+}$-Jordan arcs for the point $z_0=\infty$ and purely absolutely continuous measures belonging to the Szegő class. We extend this result in two directions. We prove explicit asymptotics of $λ_n(μ,z_0)$ for any finite measure $μ$ supported on a $C^{1+}$-Jordan arc $Γ$, and for all points $z_0\in\mathbb{C}\cup\{\infty\}\setminusΓ$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials.
We study strong asymptotics of $L^r$-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.