Chebyshev approximation of multivariable functions with the interpolation

A method of constructing a Chebyshev approximation of multivariable functions by a generalized polynomial with the exact reproduction of its values at a given points is proposed.  It is based on the sequential construction of mean-power approximations, taking into account the interpolation condition.  The mean-power approximation is calculated using an iterative scheme based on the method of least squares with the variable weight function.  An algorithm for calculating the Chebyshev approximation parameters with the interpolation condition for absolute and relative error is described.  The presented results of solving test examples confirm the rapid convergence of the method when calculating the parameters of the Chebyshev approximation of tabular continuous functions of one, two and three variables with the reproduction of the values of the function at given points.

Chebyshev approximation with the interpolation condition
multivariable functions
mean-power approximation
метод найменших квадратів
variable weight function
  1. Melnychok L. S., Popov B. A.  Best approximation of table functions with a condition.  Algorithms and programs for calculating functions on a digital computer. Institute of Cybernetics.  4, 95–102 (1977), (in Russian).
  2. Collatz L., Krabs W.  Approximationstheorie.  Tschebyscheffsche Approximation mit Anwendungen.  Teubner, Stuttgart (1973), (in German).
  3. Dunham C. B.  Discrete Chebyshev approximation with interpolation.  International Journal of Computer Mathematics.  11 (3–4), 243–245 (1982).
  4. Popov B. A., Tesler G. S.  Approximation of Functions for Engineering Applications.  Naukova Dumka, Kyiv (1980), (in Russian).
  5. Collatz L., Albrecht J.  Aufgaben aus der Angewandten Mathematik I. Gleichungen in einer oder mehreren Variablen, Approximationen.  Vieweg, Braunschweig (1972), (in German).
  6. Bomba A. J., Hladka O. M.  Problems of Identification of the Parameters of Quasiideal Filtration Processes in Nonlinear Layered Porous Media.  Journal of Mathematical Sciences. 220 (2), 213–225 (2017).
  7. Gerashchenko O. A., Gordov A. I., Eremina A. K.  Temperature measurements. Naukova Dumka, Kyiv (1989), (in Russian).
  8. Atieg A., Watson G. A.  Use of $l_{p}$ norms in fitting curves and surfaces to data.  The ANZIAM Journal.  45 (E), 187–200 (2004).
  9. Verlan A. F., Adbusadarov B. B., Ignatenko A. A., Maksimovich N. N.  Methods and devices for interpreting experimental dependencies in the study and control of energy processes. Naukova Dumka, Kyiv (1993), (in Russian).
  10. Dunham C. B.  Remez algorithm for Chebyshev approximation with interpolation.  Computing.  28, 75–78 (1982).
  11. Kondrat'ev V. P.  Uniform approximation with constraints of interpolation type.  Algorithms and programs for approximating functions.  IMM AN USSR, Sverdlovsk, 40–69 (1981), (in Russian).
  12. Dunham C., Zhu C.  Strong uniqueness of nonlinear Chebyshev approximation (with interpolation).  Numerical Mathematics and Computing. 161–169 (1990).
  13. Skopetskii V. V., Malachivskii P. S.  Chebyshev approximation of functions by the sum of a polynomial and an expression with a nonlinear parameter and endpoint interpolation.  Cybernetics and Systems Analysis.  45 (1), 58–68 (2009).
  14. Malachivskyy P. S., Skopetskii V. V.  Continuous and Smooth Minimax Spline Approximation.  Naukova Dumka, Kyiv (2013), (in Ukrainian).
  15. Korneychuk N. P., Ligun A. A., Doronin V. G.  Approximation with restrictions. Naukova Dumka, Kyiv (1982), (in Russian).
  16. Remez E. Ya.  Fundamentals of the Numerical Methods of Chebyshev Approximation.  Naukova Dumka, Kyiv (1969), (in Russian).
  17. Malachivskyy P. S., Matviychuk Y. N., Pizyur Y. V., Malachivskyi R. P.  Uniform Approximation of Functions of Two Variables.  Cybernetics and Systems Analysis.  53 (3), 426–431 (2017).
  18. Malachivskyy P. S., Pizyur Y. V., Malachivskyi R. P., Ukhanska O. M.  Chebyshev Approximation of Functions of Several Variables.  Cybernetics and Systems Analysis.  56 (1), 118–125 (2020).
  19. Malachivskyy P. S., Melnychok L. S., Pizyur Y. V.  Chebyshev Approximation of the Functions of Many Variables with the Condition.  15th International Conference on Computer Sciences and Information Technologies (CSIT). 54–57 (2020).
  20. Jukic D.  On the existence of the best discrete approximation in $l_{p}$ norm by reciprocals of real polynomials.  Journal of Approximation Theory.  156 (2), 212–222 (2009).
  21. Malachivskyy P. S., Pizyur Ya. V., Malachivskyi R. P.  Calculating the Chebyshev approximation of functions of several variables.  5th Sci.-Tech. Conf. Computational Methods and Systems of Information Transformation. 35–38 (2018), (in Ukrainian).
  22. Malachivskyy P. S., Montsibovych B. R., Pizyur Y. V., Malachivskyi R. P.  Chebyshev approximation of functions of two variables by a rational expression.  Matematychne ta Komp. Modelyuvannya.  19, 75–81 (2019), (in Ukrainian).
  23. Malachivskyy P. S., Pizyur Y. V., Malachivsky R. P.  Chebyshev Approximation by a Rational Expression for Functions of Many Variables.  Cybernetics and Systems Analysis.  56 (5), 811–819 (2020).
  24. Nakatsukasa Y., Trefethen L. N.  An Algorithm for Real and Complex Rational Minimax Approximation.  SIAM Journal on Scientific Computing.  42 (5), A3157–A3179 (2020).