# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 41A20 Approximation by rational functions 30E10 Approximation in the complex domain 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional)
Full Text:
##### 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)