The article discusses the symmetric space of periodic functions of several variables, specifically, the generalized Lorentz–Zygmund space and the Nikol’skii–Besov class within this space. Estimates for the approximation of functions from the Nikol’skii–Besov class by partial sums over step hyperbolic crosses of Fourier series are established in the uniform metric. An analog of the Jackson–Nikol’skii inequality for multiple trigonometric polynomials in the norms of the generalized Lorentz–Zygmund space and the space of continuous functions is proved.
Keywords: symmetric space, Fourier sum, Nikol’skii–Besov class, Lorentz–Zygmund space
Received August 16, 2024
Revised October 29, 2024
Accepted November 4, 2024
Funding Agency: The work was supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant AP19677486).
Gabdolla Akishev, Dr. Phys.-Math. Sci., Prof., Lomonosov Moscow University, Kazakhstan Branch, Astana, 010010 Republic of Kazakhstan; Institute of mathematics and mathematical modeling, Almaty, 050010 Republic of Kazakhstan; e-mail: akishev_g@mail.ru
REFERENCES
1. Krein S.G., Petunin Yu.I., Semenov E.M. Interpolation of linear operators. Providence, R.I.: American Math. Soc., 1982, Ser. Translations of Mathematical Monographs, vol. 54, 375 p. ISBN: 978-0821831762 . Original Russian text published in Krein S. G., Petunin Yu. I., Semenov E. M. Interpolyatsiya lineinykh operatorov, Moscow, Nauka Publ., 1978, 400 p.
2. Bennet C., Sharpley R. Interpolation of operators, NY, Acad. Press, 1988, vol. 129, 469 p. doi: 10.1016/s0079-8169(08)x6053-2
3. Semenov E.M. Interpolation of linear operators in symmetric spaces. Sov. Math., Dokl., 1965, vol. 6, pp. 1294–1298.
4. Stein E.M., Weiss G. Introduction to Fourier analysis on euclidean spaces, Princeton, Princeton Univ. Press, 1971, 312 p. ISBN: 9780691080789 . Translated to Russian under the title Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Moscow, Mir Publ., 1974, 333 p.
5. Edmunds D., Gurka P., Opic B. On embeddings of logarithmic Bessel potential spaces. J. Funct. Anal., 1997, vol. 146, iss. 1, pp. 116–150. Art. no. FU963037. doi: 10.1006/jfan.1996.3037
6. Opic B., Pick L. On generalized Lorentz–Zygmund spaces. Math. Inequal. Appl., 1999, vol. 2, no. 3, pp. 391–467. doi: 10.7153/mia-02-35
7. Bennet C., Rudnik K. On Lorentz–Zygmund spaces, Disser. Math., vol. 175, Warszawa, 1979, 66 p.
8. Lizorkin P.I., Nikol’skii S.M. Spaces of functions of mixed smoothness from the decomposition point of view. Proc. Stekov Inst. Math., 1990, vol. 187, iss. 3, pp. 163–184.
9. Amanov T.I. Prostranstva differentsiruemykh funktsii s dominiruyushchei smeshannoi proizvodnoi [Spaces of differentiable functions with dominant mixed derivative], Alma-ata, Nauka Publ., 1976, 224 p.
10. Lebesgue Н. Sur la représentation trigonometrique approchée des fonetions satisfaisant à une condition de Lipschitz. Bull. Soc. Math. France, 1910, vol. 38, pp. 184–210.
11. Oskolkov K.I. Lebesgue’s inequality in a uniform metric and on a set of full measure. Math. Notes Acad. Sci. USSR, 1975, vol. 18, pp. 895–902. doi: 10.1007/BF01153041
12. Baiborodov S.P. Lebesgue constants and approximation of functions by rectangular fourier sums in $L^p(T^m)$. Math. Notes Acad. Sci. USSR, 1983, vol. 34, pp. 522–529. doi: 10.1007/BF01160866
13. Davydov O.V. Sequences of rectangular Fourier sums of continuous functions with given majorants of the mixed moduli of smoothness. Sb. Math., 1996, vol. 187, iss. 7, pp. 981–1004. doi: 10.1070/SM1996v187n07ABEH000143
14. Stechkin S.B. The approximation of periodic functions by Fejér sums. Trudy Mat. Ins. Steklov, 1961, vol. 62, pp. 522–524 (in Russian).
15. Stečkin S.B. On the approximation of periodic functions by de la Vallée Poussin sums. Anal. Math., 1978, vol. 4, no. 1, pp. 61–74. doi: 10.1007/BF01904859
16. Dahmen W. Best approximation and de la Vallée-Poussin sums. Math. Notes Acad. Sci. USSR, 1978, vol. 23, pp. 369–376. doi: 10.1007/BF01789003
17. Geit V.É. On the exactness of certain inequalities in approximation theory. Math. Notes Acad. Sci. USSR, 1971, vol. 10, pp. 768–776. doi: 10.1007/BF01109042
18. Il’yasov N.A. On the order of approximation in the uniform metric by the Fejér–Zygmund means on the classes $E_{p}(\varepsilon)$. Math. Notes, 2001, vol. 69, no. 5, pp. 625–633. doi: 10.1023/A:1017372708394
19. Trigub R.M., Belinsky E.S. Fourier analysis and approximation of functions, Dordrecht, Springer Publ., 2004, 586 p. doi: 10.1007/978-1-4020-2876-2
20. Jia-Ding Cao. Stečkin inequalities for summability methods. Int. J. Math. Math. Sci., 1997, vol. 20, no. 1, pp. 93–100. doi: 10.1155/S0161171297000136
21. Liflyand I.R. Exact order of the Lebesgue constants of hyperbolic partial sums of multiple fourier series. Math. Notes Acad. Sci. USSR, 1986, vol. 39, pp. 369–374. doi: 10.1007/BF01156675
22. Temlyakov V.N. Estimates for the asymptotic characteristics of classes of functions with bounded mixed derivative or difference. Proc. Steklov Inst. Math., 1990, vol. 189, no. 4, pp. 161–197.
23. Romanyuk A.S. Approximability of the classes Bp,θr of periodic functions of several variables by linear methods and best approximations. Sb. Math., 2004, vol. 195, iss. 2, pp. 237–261. doi: 10.1070/SM2004v195n02ABEH000801
24. Akishev G. On the orders of M-terms approximations of classes of functions of the symmetrical space. Mat. Zh., 2014, vol. 14, no. 4, pp. 46–71 (in Russian).
25. Lapin S.V. Some embedding theorems for products of functions, Manuscript deposited in VINITI on February 20, 1985, no. 1036–80Dep., 31 p.
26. Sherstneva L.A. On the properties of best Lorentz approximations and certain embedding theorems. Sov. Math. (Iz. VUZ), 1987, vol. 31, no. 10, pp. 62–73.
27. Bekmaganbetov K.A. On orders of approximation of the Besov class in the metric of anisotropic Lorentz spaces. Ufim. math. J., 2009, vol. 1, no. 2, pp. 9–16 (in Russian).
28. Temlyakov V.N. Approximations of functions with bounded mixed derivative. Proc. Steklov Inst. Math., 1989, vol. 178, pp. 1–121.
29. Nikol’skii S.M. Approximation of functions of several variables and embedding theorems, Berlin, Heidelberg, Springer, 1975, 420 p. doi: 10.1007/978-3-642-65711-5 . Original Russian text published in Nikol’skii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Moscow, Nauka Publ., 1969, 480 p.
30. Capone C., Fiorenza A. On small Lebesgue spaces. J. Function Space Appl., 2005, vol. 3, no. 1, pp. 73–89. doi: 10.1155/2005/192538
31. Kokilashvili V., Meskhi A., Rafeiro H., Samko S. Integral operators in non-standard function spaces. Vol. 1 : Variable exponent Lebesgue and amalgam spaces, Cham, Birkhäuser, 2016, 567 p. doi: 10.1007/978-3-319-21015-5
32. Neves J.S. Lorentz–Karamata spaces, Bessel and Riesz potential and embeddings. Disser. Math., 2002, vol. 405, pp. 1–46. doi: 10.4064/dm405-0-1
33. Temlyakov V.N. On optimal recovery in $L_2$. J. Complexity, 2021, vol. 65, art. no. 101545. doi: 10.1016/j.jco.2020.101545
34. Krieg D., Pozharska K., Ullrich M., Ullrich T. Sampling recovery in the uniform norm [e-resource]. 2023. 29 p. https://doi.org/10.48550/arXiv.2305.07539
Cite this article as: G. Akishev. On estimates of the approximation of functions from a symmetric space by Fourier sums in the uniform metric. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 9–26.
