n-Widths, Faber Expansion, and Computation of Analytic Functions

AbstractLetBH∞(Ω) be the space of analytic functionsfin the region Ω for which |f(z)| ≤ 1,z∈ Ω, and letKbe a compact subset of Ω. How can we compute the values of any functionf∈BH∞(Ω) at an arbitrary pointz∈K? One of the approaches to this problem applies the results concerning then-widths and ϵ-entrophy of classBH∞(Ω) in the metricC(K). In the case whenKhas a simply connected complement inC and Ω is a canonical neighbourhood ofK, the classical tools for approximation off∈BH∞(Ω) inC(K) give the Faber series. This work is concerned with the following: the exact values of Kolmogorov and othern-widths of Hardy spacesHp, then-widths and ϵ-entrophy of classBH∞(Ω), the optimality of Faber approximations, and computing values of analytic functions with the help of Faber series