Existence and explicit form of nonlinear Hermite–Chebyshev approximations
Abstract
In this paper, sufficient conditions for the existence of trigonometric Hermite–Jacobi approximations of a system of functions that are sums of convergent Fourier series are found. Based on these results, sufficient conditions are established under which nonlinear Hermite–Chebyshev approximations of systems of functions representable by Fourier series in Chebyshev polynomials of the first and second kind exist. When the found conditions are met, explicit formulas are obtained for the numerators and denominators of trigonometric Hermite–Jacobi approximations and nonlinear Hermite–Chebyshev approximations of the first and second kind of the specified systems of functions.
About the Authors
A. P. Starovoitov
F. Skorina Gomel State University
Belarus
Belarus
Gomel
I. V. Kruglikov
F. Skorina Gomel State University
Belarus
Belarus
Gomel
References
1. Nikishin E. M., Sorokin V. N. Rational approximations and orthogonality. Moscow, Nauka, 1988 (in Russian).
2. Suetin S. P. On the existence of nonlinear Pade´–Chebyshev approximations for analytical functions. Mathematical Notes, 2009, vol. 86, iss. 2, pp. 290–303 (in Russian).
3. Gonchar A. A., Rakhmanov E. A., Suetin S. P. Pade´–Chebyshev approximations for multivalued analytical functions, equilibrium energy variation and the S-property of stationary compacts. Russian Mathematical Surveys, 2011, vol. 66, iss. 6, pp. 3–36 (in Russian).
4. Baker Jr. G., Graves-Morris P. Pade´ Approximations. 1. Fundamentals of theory. 2. Generalizations and applications. Moscow, Mir, 1986 (in Russian).
5. Labych Yu. A., Starovoitov A. P. Trigonometric Pade´ approximations of functions with regularly decreasing Fourier coefficients. Sbornik: Mathematics, 2009, vol. 200, iss. 7, pp. 107–130 (in Russian).
6. Geddes K. O. Block structure in the Chebyshev–Pade´ table. SIAM J. Numer. Anal., 1981, vol. 18, iss. 5, pp. 844–861.
7. Adukov V. M., Ibryaeva O. L. The asymptotic behavior of the denominators of the Pade´– Chebyshev approximations for the last intermediate row. The rational case. Bulletin of the South Ural State University, series «Mathematics. Mechanics. Chemistry», 2005, vol. 6, iss. 6, pp. 11–18 (in Russian).
8. Ibryaeva O. L. Sufficient condition for the uniqueness of the linear Pade´–Chebyshev approximation. News of the Chelyabinsk Scientific Center, 2002, iss. 4, pp. 1–5 (in Russian).
9. Starovoitov A. P., Kechko E. P., Osnach T. M. On the existence of trigonometric Hermite–Jacobi approximations and nonlinear Hermite–Chebyshev approximations. Journal of the Belarusian State University. Mathematics and Informatics, 2023, iss. 2, pp. 6–17 (in Russian).
10. Starovoitov A. P., Kechko E. P., Osnach T. M. Existence and uniqueness of consistent Hermite– Fourier approximations. Problems of Physics, Mathematics and Technics, 2023, iss. 2 (55), pp. 68–73 (in Russian).
11. Starovoitov A. P., Kruglikov I. V., Osnach T. M. Rational approximations of power series, trigonometric series and series of Chebyshev polynomials. Journal of the Belarusian State University. Mathematics and Informatics, 2025, vol. 61, iss. 3, pp. 6–21 (in Russian).
12. Mason J. C., Crampton A. Laurent–Pade´ approximants to four kinds of Chebyshev polynomial expansions. I. Maehly type approximants. Numer. Algorithms, 2005, vol. 38, pp. 3–18.
13. Mason J. C., Crampton A. Laurent–Pade´ approximants to four kinds of Chebyshev polynomial expansions. II. Clenshaw–Lord type approximants. Numer. Algorithms, 2005, vol. 38, pp. 19–29.
14. Suetin S. P. Pade´ approximations and effective analytical continuation of the power series. Russian Mathematical Surveys, 2002, vol. 57, iss. 1, pp. 45–142 (in Russian).
15. Suetin S. P. Questions of convergence of the Pade´–Faber approximations. Ph. D. thesis. Moscow: MGU, 1981 (in Russian).
16. Starovoitov A. P., Kruglikov I. V. Uniqueness and explicit form of linear Hermite–Chebyshev approximations. Problems of Physics, Mathematics and Technics, 2025, iss. 1(62), pp. 67–72 (in Russian).
17. Hermite C. Sur la fonction exponentielle. Paris, C. R. Akad. Sci., 1873, vol. 77, pp. 18–293.
18. Starovoitov A. P., Ryabchenko N. V. On determinant representations of Hermite–Pade´ polynomials. Transactions of the Moscow Mathematical Society, 2022, vol. 83, iss. 1, pp. 17–35 (in Russian).
19. Starovoitov A. P., Ryabchenko N. V. Uniqueness of the solutions of the Hermite–Pade´ problems. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series, 2019, vol. 55, iss. 4, pp. 445–456 (in Russian).
20. Jacobi C. Uber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function. J. Reine Angew. Math., 1846, vol. 30, pp. 127–156.
21. Aptekarev A. I., Buslaev V. I., Martinez–Finkelshtein A., Suetin S. P. Pade´ approximants, continued fractions, and orthogonal polynomials. Russian Mathematical Surveys, 2011, vol. 66, iss. 6 (402), pp. 37–122 (in Russian).
22. Osnach T. M., Ryabchenko N. V., Starovoitov A. P. An analogue of Jacobi’s theorem for simultaneous Hermitian interpolation of several functions. Problems of Physics, Mathematics and Technics, 2023, iss. 1 (54), pp. 89–92 (in Russian).
23. Németh G., Páris G. The Gibbs phenomenon in generalized Pade´ approximation. J. Math. Phys., 1985, vol. 26, iss. 6, pp. 1175–1178.
24. De Bruin M. G. Convergence of the Pade´ table for <sub>1</sub>F<sub>1</sub>(1; c; x) K. Nederl. Akad. Wetensch., Ser. A., 1976, vol. 79, pp. 408–418.
25. Aptekarev A. I. On approximations of the Pade´ to the set {<sub>1</sub>F<sub>1</sub>(1, c; λ<sub>i</sub>z)}<sup>k</sup><sub>i=1</sub>. Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 1981, iss. 2, pp. 58–62 (in Russian).
26. Starovoitov A. P. Hermite–Pade´ approximations of Mittag–Leffler functions. Proceedings of the Steklov Institute of Mathematics, 2018, vol. 301, pp. 241–258 (in Russian).
27. Andrianov I. V. Application of Pade´ approximants in perturbation methods. Adv. in Mech., 1991, vol. 14, iss. 2, pp. 3–25.
28. Andrianov I. V., Awrejcewicz J. New trends in asymptotic approaches: summation and interpolation methods. Appl. Mech. Rev., 2001, vol. 54, iss. 1, pp. 69–92.
29. Knizhnerman L. A. Identification of potential field poles using Fourier–Chebyshev series expansion. Proceedings of the USSR Academy of Sciences. Earth Physics Series, 1984, iss. 11, pp. 119–123 (in Russian).
30. Ermokhin K. M. Continuation of geophysical fields into the area of anomaly sources by the method of approximation by chain fractions. Geophysics, 2007, iss. 1, pp. 51–55 (in Russian).
31. Prokhorov G. V., Kolbeev V. V., Zhelnov K. I., Ledenev M. A. Maple V Release 4 Math Package: User’s Guide. Kaluga: Oblizdat, 1998 (in Russian).
32. Tee T. W., Trefethen L. N. A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput., 2006, vol. 28, iss. 5, pp. 1798–1811.
Review
For citations:
Starovoitov A.P., Kruglikov I.V. Existence and explicit form of nonlinear Hermite–Chebyshev approximations. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(1):75-86. (In Russ.)
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