Iryna Victorivna Brysina, Victor Olexandrovych Makarichev


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


data processing; data compression; atomic functions; up-function; Fup-function; generalized Fup-function; discrete atomic transform; generalized discrete atomic transform

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Rvachev, V. L., Rvachev, V. A. Neklassicheskie metody teorii priblizhenii v kraevykh zadachakh [Nonclassical methods of approximation theory in boundary value problems]. Kyiv, “Naukova dumka” Publ., 1979. 196 p.

Rvachev, V. A. Compactly supported solutions of functional-differential equations and their applications. Russian Math. Surveys, 1990, vol. 45, no. 1, pp. 87-120.

Rvachev, V. A. On approximation by means of the function up(x). Sov. Math. Dokl. 1977, vol. 233, no. 2, pp. 295-296.

Brysina, I. V., Makarichev, V. A. Generalized atomic wavelets. Radioelectronic and Computer Systems, 2018, vol. 85, no. 1, pp. 23-31.

Stoyan, Yu. G., Protsenko, V. S., Man’ko, G. P., Goncharyuk, I. V., Kurpa, L. V., Rvachev, V. A., Sinekop, N. S., Sirodzha, I. B., Shevchenko, A. N., Sheiko, T. I. Teorija R-funkcij i aktual’nye problemy prikladnoj matematiki [Theory of R-functions and current problems of applied mathematics]. Kyiv, “Naukova dumka” Publ., 1986. 264 p.

Makarichev, V. A. Approximation of periodic functions by mups(x). Math. Notes, 2013, vol. 93, no. 6, pp. 858-880.

Brysina, I. V., Makarichev, V. A. On the asymptotics of the generalized Fup-functions. Adv. Pure Appl. Math., 2014, vol. 5, no. 3, pp. 131-138.

Brysina, I. V., Makarichev, V. A. Approximation properties of generalized Fup-functions. Visnyk of V. N. Karazin Kharkiv National University, Ser. “Mathematics, Applied Mathematics and Mechanics”, 2016, vol. 84, pp. 61-92.

Makarichev, V. A. Ob odnoi nestatsionarnoi sisteme beskonechno differentsiruemykh veievletov s kompaktnym nositelem [On the nonstationary system of infinitely differentiable wavelets with a compact support]. Visnyk KhNU, Ser. “Matematika, prikladna matematika and meckhanika”, 2011, no. 967, pp. 63-80.

Brysina, I. V., Makarichev, V. A. Atomic wavelets. Radioelectronic and Computer Systems, 2012, vol. 53, no. 1, pp. 37-45.

Makarichev, V. A. The function mups(x) and its applications to the theory of generalized Taylor series, approximation theory and wavelet theory. Contemporary problems of mathematics, mechanics and computing sciences, Kharkiv, “Apostrophe” Publ., 2011, pp. 279-287.

Kuan-Ching Li, Hai Jiang, Zomaya, A. Y. (eds.). Big data management and processing, Chapman and Hall / CRC, 2017. 487 p.

Gonzalez, R. C., Woods, R. E. Digital image processing, Prentice Hall, 2008. 977 p.

Salomon, D., Motta, G., Bryant, D. Handbook of data compression, Springer, 2010, 1370 p.

Tsay, R. S. Analysis of financial time series, John Wiley and Sons, 2010, 714 p.

Makarichev, V. O. Application of atomic functions to lossy image compression.Theoretical and applied aspects of cybernetics.Proceedings of the 5th International scientific conference of students and young scientists, Kyiv, “Bukrek” Publ., 2015, pp. 166-175.



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