On approximation of non-analytic functions by analytical ones

We study the properties of the elements of best approximation for functions summed up over the unit circle of functions by functions from the Bergman space. For approximable functions of a special type, we five a sufficiently accurate description of the properties of these elements in terms of the Hardy and Lipschitz classes. The result obtained is based on an analysis of the corresponding duality relation for extremal problems. The developed method is also applicable to relatively smooth (in terms of Sobolev spaces) approximable functions.