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A note on rational $L^p$ approximation on Jordan curves. (English) Zbl 1280.41015
The author starts with a nice introduction into the subject at hand: Given a rectifiable Jordan curve $T$ with interior domain $G$ and exterior domain $O$ (with respect to $\overline{\text{\bf C}}$), the set $A(G)$ consists of functions $f$ with {\parindent=6mm \item{$\bullet$} $f$ vanishes at infinity and admits holomorphic and single-valued continuation from infinity to an open neighborhood of $\overline{O}$, \item{$\bullet$} $f$ admits meromorphic, possibly multi-valued, continuation along any arc in $G\setminus E_f$ starting from $T$, where $E_f$ is a finite set of points in $G$, \item{$\bullet$} $E_f$ is non-empty, the meromorphic continuation of $f$ from infinity has a branch point at each element of $E_f$. \par} The reader is referred to the important paper by {\it L. Baratchart} et al. [Adv. Math. 229, No. 1, 357--407 (2012; Zbl 1232.41014)], where the theory is developed and which contains the a.o. result $$\lim_{n\rightarrow\infty}\, \rho_{n,p}^{1/2n}(f,O)=\exp{\left(-{1\over\text{cap}(K_T,T)}\right)}\tag1$$ for $p=2$ ($\rho_{n,p}(f,O)$ is the error of the best approximation in $L^{p}(s_T)$, approximation is done with rational functions $p/q$ with deg$\,p=n-1$, deg$\,q=n$, $s_T$ is the arc measure on $T$). As (1) follows for $p=\infty$ from a result in a paper by {\it A. A. Gonchar} and {\it E. A. Rakhmanov} [Math. USSR, Sb. 62, No. 2, 305--348 (1989); translation from Mat. Sb., Nov. Ser. 134(176), No. 3(11), 306--352 (1987; Zbl 0663.30039)], it holds for all $2\leq p\leq \infty$. In the paper under review, (1) is derived for all $1\leq p\leq\infty$, directly from Theorem 1 in the paper by Gonchar and Rakhmanov cited above.
MSC:
41A20Approximation by rational functions
30E10Approximation in the complex domain
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
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References:
[1] Baratchart, L., Stahl, H., Yattselev, M.: Weighted extremal domains and best rational approximation. Adv. Math. 2012, 357--407 (2012) · Zbl 1232.41014 · doi:10.1016/j.aim.2011.09.005
[2] Gonchar, A.A., Rakhmanov, E.A.: Equilibrium distributions and the degree of rational approximation of analytic functions. Mat. Sb. (Russ.) 134(176), 306--352 (1987); English Transl. in Math. USSR Sb. 62, 305--348 (1989) · Zbl 0645.30026
[3] Pommerenke, Ch.: Boundary Behavior of Conformal Mappings, Grundlehren der mathematischen Wissenschaften, 299. Springer, Berlin (1992)