Trigonometric transformation of functions of several real variables in the study and approximation of functions

Aim. Defining the transformation of a set of functions of two real variables based on the concept of a directional derivative, representing methods for applying the transformation to explore functions and their approximation, constructing numerical methods and determining the transformation for functions of more variables.

Methodology. Analysis of the theory of sets and their mappings, including mappings of a set of functions, the use of relevant sections of the theory of mathematical analysis, including the transformation of functions and the expansion of non-periodic and periodic functions into series, the theory of constructing numerical methods for approximating functions.

Results. The transformation of functions of two real variables is defined, methods of applying the transformation to study functions in the vicinity of a given point and expansion of functions into special functional series are described, centrifugally periodic functions of two variables are defined, numerical methods for approximating functions of two variables are constructed, and transformation of functions of a larger number of variables is defined.

Research implications. The presented transformation of functions of several variables is an element of the further development of the theory of functions and functional analysis; the described methods of its use expand the directions for research of this theory, including the area of expansion of functions of several variables into functional series. The theoretical results obtained can be used in applied problems of mathematical modeling, including the use of centrifugally periodic functions for modeling wave processes (in seismology, explosion physics and other fields), in approximating functions using the presented numerical methods, and in the development of new numerical methods.

1. Ilyin, V. A., Sadovnichy, V. A. & Sendov, B. Kh. (1985). Mathematical Analysis. Basic Course. Moscow: Lomonosov Moscow State University publ. (in Russ.).

2. Ulyanov, A. P. (2019). Fundamentals of Mathematical Analysis for Physics Students. Part 3. Functions of Several Variables. Novosibirsk: Novosibirsk State University publ. (in Russ.).

3. Ditkin, V. A. & Prudnikov, A. P. (1961). Integral Transforms and Operational Calculus (Series "Reference Mathematical Library"). Moscow: Fizmatgiz publ. (in Russ.).

4. Volkov, V. A. (2014). Fourier Series. Fourier and Radon Integral Transforms. Yekaterinburg: Ural University publ. (in Russ.).

5. Fadeev, M. A. & Markov, K. A. (2010). Numerical Methods. N. Novgorod: UNN publ. (in Russ.).

6. Arushanyan, I. O. (2017). Algorithms for Unconstrained Minimization of Functions of Many Variables. Moscow: Lomonosov Moscow State University publ. (in Russ.).

7. Smirnov, A. V. (2025). Method for estimating objective function landscape convexity during extremum search. In: Russian Technological Journal, 13 (2), 121–131. DOI: 10.32362/2500-316X-2025-13-2-121-131 (in Russ.).

8. Bakhtiyarova, L. I. (2017). Numerical methods for solving the problem of finding a convex spatial figure of revolution of maximum surface area under given restrictions on its width. In: Young scientist, 9 (143), 1–7 (in Russ.).

9. Valger, S. A., Fedorova, N. N. & Fedorov, A. V. (2017). Mathematical modeling of propagation of explosion waves and their effect on various objects. In: Combustion, Explosion, and Shock Waves, 53 (4), 72–83. DOI: 10.1134/S0010508217040074.

10. Skobel'tsyn, S. A. & Tolokonnikov, L. A. (2019). Modelling of sound scattering by sphere with an inhomogeneous coating in a plane waveguide. In: Izvestiya Tula State University. Technical sciences, 9, 150–158.

11. Stepanov, P. Yu. & Gomanyuk, J. A. (2022). Mathematical modeling of seismic wave kinematics in complex media. In: Moscow University Geology Bulletin, 6, 167–178. DOI: 10.33623/0579-9406-2022-6-167-178 (in Russ.).

12. Nenashev, V. A. & Blaunstein, N. S. (2016). Modeling Radio Wave Propagation in an Earth-Atmosphere Channel. In: Information and Control Systems, 6, 25–33. DOI: 10.15217/issn1684-8853.2016.6.25.