Iryna Victorivna Brysina, Victor Olexandrovych Makarichev


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


data processing; wavelets; atomic functions; V. A. Rvachev up-function; atomic wavelets; generalized atomic wavelet expansion

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Novikov, I. Ya., Stechkin, S. B. Basic wavelet theory. Russian Math. Surveys, 1990, vol. 53, no. 6, pp. 1159-1231.

Welstead, S. Fractal and wavelet image compression techniques. SPIE Press, 1999. 256 p.

Meyer, F. G., Petrossian, A. A. Wavelets in signal and image analysis. Springer, 2001. 556 p.

Stollnitz, E. J., DeRose, T. D., Salesin, D. H. Wavelets for computer graphics: theory and applications. Morgan Kaufmann Publ., 1996. 246 p.

Cohen, J., Zayed, A. I. (eds.). Wavelets and multiscale analysis: theory and applications. Springer, 2011. 353 p.

Hramov, A. E., Koronovsky, A. A., Makarov, V. A., Pavlov, A. N., Sitnikova, E. Wavelets in Neuroscience. Springer, 2015. 331 p.

Chandrasekhar, E., Dimri, V. P., Gadre, V. M. Wavelets and fractals in Earth system sciences. CRC Press, 2014. 294 p.

Farouk, M. H. Application of wavelets in speech processing. Springer, 2014. 53 p.

Chan, A. K., Goswavi, J. C. Fundamentals of wavelets: theory, algorithms and applications. John Wiley and sons, 2011. 359 p.

Gencay, R., Selcuk, F., Whitcher, B. An introduction to wavelets and other filtering methods in finance and economics. Academic press, 2002. 359 p.

Gallegati, M., Semmler, W. (eds.). Wavelet applications in economics and finance. Springer, 2014. 261 p.

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.

Spiridonov, V. Vsplesk revolyutsii [Splash of revolutions]. Available at: /1998/236/193919/ (accessed 12.01.2018).

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

Gotovac, H., Cvetkovic, V., Andricevic, R. Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: methodology, accuracy and convergence. Adv. Water Resour., 2009, vol. 32, no. 6, pp. 885-905.

Gotovac, H., Andricevic, R., Gotovac, B. Multi-resolution adaptive modeling of groundwater flow and transport problems. Adv. Water Resour., 2007, vol. 30, vo. 5, pp. 1105-1126.

Lazorenko, O. V. The use of atomic functions in the Choi-Williams analysis of ultrawideband signals. Radioelectronics and Communications Systems, 2009, vol. 52, pp. 397-404.

Ulises Moya-Sanchez, E., Bayro - Corrocha-no, E. Quaternionic analytic signal using atomic functions. Porgress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Note in Computer Science., 2012, vol. 7441, pp. 699-706.

Dyn, N., Ron, A. Multiresolution analysis by infinitely differentiable compactly supported functions. Appl. Comput. Harmon. Anal., 1995, vol. 2, no. 1, pp. 15-20.

Cooklev, T., Berbecel, G. I., Venetsanopou-los, A. N. Wavelets and differential-dilatation equations. IEEE Transactions on signal processing, 2000, vol. 48, no. 8, pp. 670-681.

Charina, M., Stockler, J. Tight wavelet frames for irregular multiresolution analysis. Appl. Comput. Harmon. Anal., 2008, vol. 25, no. 1, pp. 98-113.

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

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. Radioelektronni i komp'uterni sistemi - 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.

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.

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.

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.



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