Self-similar structures and fractal transforms in approximation theory

An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of quantum physics, many-body physics, physics of complex systems, and various aspects of applied physics and applied mathematics. A general approach for dealing with such problems has been developed, called self-similar approximation theory. A concise survey of the main ideas of this approach is presented, with the emphasis on the basic notion of group self-similarity. The techniques are illustrated by examples from quantum theory.