GENERALIZED ATOMIC WAVELETS

The problem of big data sets processing is considered. Efficiency of algorithms depends mainly on the appropriate mathematical tools. Now there exists a wide variety of different constructive tools for information analysis. Atomic functions are one of them. Theory of atomic functions was developed by V. A. Rvachev and members of his scientific school. A number of results, which prove that application of atomic functions is reasonable, were obtained. In particular, atomic functions are infinitely differentiable. This property is quite useful for smooth data processing (for example, color photos). Also, these functions have a local support, which allows to decrease complexity of numerical algorithms. Besides, it was shown that spaces of atomic functions have good approximation properties, which can reduce the error of computations. Hence, application of atomic functions is perspective. There are different ways to use atomic functions and their generalizations in practice. One such approach is a construction and application of wavelet-like structures. In this paper, generalized atomic wavelets are constructed using generalized Fup-functions and formulas for their evaluation are obtained. Also, the main properties of generalized atomic wavelets are presented. In addition, it is shown that these wavelets are smooth functions with a local support and have good approximation properties. Furthermore, the set of generalized atomic wavelets is a wide class of functions with flexible parameters that can be chosen according to specific needs. This means that the constructive analysis tool, which is introduced in this paper, gives researches and developers of algorithms flexible possibilities of adapting to the specifics of various problems. In addition, the problem of representation of data using generalized atomic wavelets is considered. Generalized atomic wavelets expansion of data is introduced. Such an expansion is a sum of trend or principal value function and several functions that describe the corresponding frequencies. The remainder term, which is an error of approximation of data by generalized atomic wavelets, is small. To estimate its value the inequalities from the previous papers of V. A. Rvachev, V. O. Makarichev and I. V. Brysina can be used