Some Fundamental Problems in the Approximate and Exact Representation of Functions of One or Several Variables

The difficulty of indicating to within e a function f belonging to a class F can be considered from the standpoint of the “amount of information” contained in the indication. In this approach, a natural characteristic of the class F is the function $$I_{F}^{a}\left( \varepsilon \right) = \log N_{F}^{a}\left( \varepsilon \right),$$ Where \(N_F^a\left( \varepsilon \right)\) is the minimum number of points in an c-net in F.This paper reviews earlier published and recently obtained estimates for the rate of growth of the function \(I_F^a\left( \varepsilon \right),\) as \(\varepsilon \to 0,\) for some classes of analytic functions and functions possessing a given number of derivatives.