Self-similar structures and fractal transforms in approximation theory

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 wit...

Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

Theory of Generalized Fractal Transforms

The most popular \fractal-based" algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and...

Generalized fractal transforms and self-similar objects in cone metric spaces

We use the idea of a scalarization of a cone metric to prove that the topology generated by any cone metric is equivalent to a topology generated by a related metric. We then analyze the case of an ordering cone with empty interior and we provide alternative definitions based on the notion of interior cone. Finally we discuss the implications of such cone metrics in the theory of iterated funct...

Statistically Self - Similar Fractal Sets

In the present paper we define statistically self-similar sets, and, using a modification of method described in [2], [3], find a Hausdorff dimension of a statistically self-similar set.