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On the best linear approximation of holomorphic functions. (English. Russian original) Zbl 1355.30035

J. Math. Sci., New York 218, No. 5, 678-698 (2016); translation from Fundam. Prikl. Mat. 19, No. 5, 185-212 (2014).
Let \(\Omega\) be an open subset of the complex plane \(\mathbb C\) and \(E\) be a compact subset of \(\Omega\). The space \(H^{\infty}(\Omega)\) consists of all bounded analytic functions in \(\Omega\). Linear methods of approximation to the class \(H^{\infty}(\Omega)\) in the space \(C(E)\) and some related problems involving the linear \(n\)-width, the Kolmogorov \(n\)-width and the \(\varepsilon\)-entropy of classes of Hardy-Sobolev type in \(L^{p}\)-spaces, \(1\leq p\leq\infty\), are discussed. Generalizations of Faber series are given for the case where \(\Omega\) is a multiply connected domain or a disjoint union of several such domains, and \(E\) is the union of a finite number of pairwise disjoint bounded continua, with some applications in numerical analysis. Multivariate analogues of some results of Babenko, Tikhomirov, Taikov and Pinkus on the approximation of analytic functions with bounded derivatives are obtained. Also, the values of \(n\)-widths and asymptotic formulas for the \(\varepsilon\)-entropy of classes of holomorphic functions in symmetric domains of tube type having bounded fractional derivatives are investigated.

MSC:

30E10 Approximation in the complex plane
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A30 Other generalizations of function theory of one complex variable
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
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