DOI: https://doi.org/10.15588/1607-3274-2019-2-5

### POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR FRACTAL PROCESSES

V. N. Gorev, A. Yu. Gusev, V. I. Korniienko

#### Abstract

Context. We consider a Kolmogorov-Wiener filter for fractal random processes, which, for example, may take place in modern
information-telecommunication systems and in control of complex technological processes. The weight function of the considered
filter may be applied to data forecast in the corresponding systems.
Objective. As is known, in the continuous case the Kolmogorov-Wiener filter weight function obeys the Fredholm integral
equation of the first kind. The aim of the work is to obtain the Kolmogorov-Wiener filter weight function as an approximate solution
of the corresponding integral equation.
Method. The truncated orthogonal polynomial expansion method for approximate solution of the Fredholm integral equation of
the first kind is used. A set of orthonormal polynomials is used.
Results. We obtained approximate results for the Kolmogorov-Wiener weight function for fractal processes with a power-law
structure function. The weight function is found as an approximate solution of the Fredholm integral equation of the first kind the
kernel of which is the correlation function of the corresponding fractal random process. Analytical results for the one-, two-, three-,
four- and five-polynomial approximations are obtained. A numerical comparison of the left-hand and right-hand sides of the integral
equation for the obtained weight functions is given for different values of the parameters. The corresponding numerical investigation
is made up to the 18-polynomial approximation on the basis of the Wolfram Mathematica package. The applicability of the obtained
solutions is discussed.
Conclusions. The Kolmogorov-Wiener weight function for fractal processes is obtained approximately in the form of a truncated
orthogonal polynomial series. The validity of the obtained weight functions is discussed. The obtained results may be applied to the
data forecast in a wide variety of different systems where fractal random processes take place.

#### Keywords

Kolmogorov-Wiener filter weight function; truncated orthogonal polynomial expansion; Fredholm integral equation of the first kind; approximate solution

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#### References

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#### GOST Style Citations

1. Method and algorithms of nonlinear dynamic processes identification / [V. Korniienko, S. Маtsyuк, I. Udovik, A. Alekseyev] // Scientific Bulletin of National Mining University. – 2016. – No. 1. – P. 98–103.
2. Kornienko V. Methods and principles of control over the complex objects of mining and metallurgical production / V. Kornienko, A. Gerasina, A. Gusev // In book “Energy Efficiency Improvement of Geotechnical Systems”. – Taylor & Francis Group, London, 2013. – P. 183–192. ISBN 978-1-138-00126-8.
3. Bagmanov V. Kh. Prognozirovanie teletraffika na osnove fraktalnykh filtrov / V. Kh. Bagmanov, A. M. Komissarov,
A. Kh. Sultanov // Vestnik Ufimskogo gosudarstvennogo aviatsionnogo universiteta. – 2007. – Vol. 9, No. 6 (24). – P. 217–222.
4. Pipiras V. Long-Range Dependence and Self-Similarity / V. Pipiras, M. Taqqu. – Cambridge University Press, 2017. – 668 p. DOI: 10.1017/CBO9781139600347
5. Gorev V. N. On the analytical solution of a Volterra integral equation for investigation of fractal processes / V. N. Gorev,
A. Yu. Gusev, V. I. Korniienko // Radio Electronics, Computer Science, Control. – 2018. – No. 4. – P. 42–50.
6. Miller S. Probability and Random Processes With Applications to Signal Processing and Communications. Second edition /
S. Miller, D. Childers. – Amsterdam: Elseiver/Academic Press, 2012. – 598 p. DOI: doi.org/10.1016/B978-0-12-386981-4.50001-1
7. Gorev V. N. Plasma kinetic coefficients with account for relaxation processes / V. N. Gorev, A. I. Sokolovsky // International Journal of Modern Physics B. – 2015. – Vol. 29, No. 32. – P. 1550233 (23 pages).
8. Loyalka S. K. Chapman-Enskog solutions to arbitrary order in Sonine polynomials I: Simple, rigid-sphere gas / S. K. Loyalka,
E. L. Tipton, R. V. Tompson // Physica A. – 2007. – Vol. 379. – P. 417–435.
9. Polyanin A. D. Handbook of the integral equations. Second edition / A. D. Polyanin, A. V. Manzhirov. – Boca Raton:
Chapman & Hall/CRC Press. – Taylor & Francis Group. – 2008. – 1143 p.
10. Ismail Mourad E. H. Classical and Quantum Orthogonal Polynomials in One Variable / E. H. Ismail Mourad. – Cambridge University Press, 2005. – 708 p.
11. Ziman J. M. Electrons and Phonons. The Theory of Transport Phenomena in Solids / J. M. Ziman. – Oxford University Press, 2001. – 576 p.

Copyright (c) 2019 V. N. Gorev, A. Yu. Gusev, V. I. Korniienko