ATOMIC FUNCTIONS AND THEIR GENERALIZATIONS IN DATA PRO-CESSING: FUNCTION THEORY APPROACH

Abstract

Theory of atomic functions, which are solutions with a compact support of the linear functional differential equations with a constant coefficients and linear transforms of the argument, was created in the 70's of the 20th century because of the necessity to solve different applied problems, in particular, boundary value problems. One of the reasons for the appearance of atomic functions and some other classes of functions was the inability to use such classic approximation tools as algebraic and trigonometric polynomials. V.A. Rvachev up-function is the most famous and widely used atomic function. With the passage of time and the development of technologies, the existing problems are changing and fundamentally new problems appear. For instance, now the big data processing is one of the most important problems. It should be mentioned that the suitable mathematical tools must be used to obtain the desired result. This paper is devoted to fundamentals of applications of some atomic functions and their generalizations to data processing and lossy information compression. In this paper we consider the main properties of these functions from the function theory point of view and give their interpretation with respect to information processing. Smoothness, compact support and good approximation properties are the main advantages of atomic functions. Moreover, the spaces of atomic functions and the spaces of generalized Fup-functions, which are the natural generalization of V.A. Rvachev Fup-functions, are asymptotically extremal for approximation of periodic differentiable functions. This means that in the terms of A.N. Kolmogorov width these functions are just as effective as classic trigonometric polynomials {1, cos(nx), sin(nx)}. Hence, the replacement of discrete transforms based on trigonometric functions on similar transforms based on atomic functions and generalized Fup-functions is quite promising. For this purpose we introduce discrete atomic transform and generalized discrete atomic transform. We also discuss the dependence of data processing results on order of smoothness and size of support of the used functions. Theoretical justification of the application of some atomic functions and generalized Fup-functions to data processing and, in particular, lossy data compression is the main result of this paper