G. Akishev. An inequality of different metrics in the generalized Lorentz space ... P. 5-18

The main goal of the paper is to prove the Jackson-Nikol'skii inequality for multiple trigonometric polynomials in the generalized Lorentz space $L_{\psi,\theta}(\mathbb{T}^{m})$.  In the first section we give definitions of a symmetric space of functions, a fundamental function, and the Boyd index of a space. In particular, we define the generalized Lorentz and Lorentz-Zygmund spaces. In addition, definitions of a weakly varying function and of the Lorentz-Karamata space are given. In the second section we prove an analog of the inequality of different metrics for multiple trigonometric polynomials in generalized Lorentz spaces $L_{\psi,\theta}(\mathbb{T}^{m})$ with identical Boyd indices but different fundamental functions. In the Lorentz-Karamata space, the order-exact Jackson-Nikol'skii inequality for multiple trigonometric polynomials is obtained.

Keywords: Lorentz-Karamata space, Jackson-Nikol'skii inequality, trigonometric polynomial

Received August 29, 2018

Revised November 23, 2018

Accepted November 26, 2018

Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Gabdolla Akishev, Dr. Phys.-Math. Sci., Prof., L.N.Gumilyov Eurasian National University, 100008, Astana , Republic Kazakhstan; Ural Federal University, Yekaterinburg, 620002 Russia,
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 Mathematical Society, 1982, Ser. Translations of Mathematical Monographs, vol. 54, 375 p. Original Russian text published in Krein S.G., Petunin Yu.I., Semenov E.M., Interpolyatsiya lineinykh operatorov, Moscow: Nauka Publ., 1978, 400 p.

2.   Stein E.M. and Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ. Press, 1971, 312 p. ISBN: 9781400883899 . Translated to Russian under the title Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh. Moscow, Mir Publ., 1974, 332 p.

3.   Semenov E.M. Interpolation of linear operators in symmetric spaces. Sov. Math., Dokl., 1965, vol. 6, pp. 1294–1298.

4.   Jackson D. Certain problems of closest approximation. Bull. Amer. Math. Soc., 1933, vol. 39, no. 12, pp. 889–906. doi: 10.1090/S0002-9904-1933-05759-2 .

5.   Nikol’skii S.M. Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables. In: Azarin V.S. et al (eds.), Thirteen papers on functions of real and complex variables. Providence, R.I.: Amer. Math. Soc., 1969, pp. 1–38, ISBN: 978-1-4704-3291-1 .

6.   Bari N.K. Generalization of inequalities of S.N. Bernstein and A.A. Markov. Izvestiya Akad. Nauk SSSR. Ser. Mat., 1954, vol. 18, no. 2, pp. 159–176 (in Russian).

7.   Ivanov V.I. Certain inequalities in various metrics for trigonometric polynomials and their derivatives. Math. Notes, 1975, vol. 18, no. 4, pp. 880–885. doi: 10.1007/BF01153038 .

8.   Arestov V.V. Inequality of different metrics for trigonometric polynomials. Math. Notes, 1980, vol. 27, no. 4, pp. 265–269. doi: 10.1007/BF01140526 .

9.   Arestov V.V., Deikalova M.V. Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere. Proc. Steklov Inst. Math., 2014, vol. 284, suppl. 1, pp. 9–23. doi: 10.1134/S0081543814020023 

10.   Temlyakov V.N. Approximation of functions with a bounded mixed derivative. Proc. Steklov Inst. Math., 1989, vol. 178, pp. 1–121.

11.   Rodin V.A. Jackson and Nikol’skii inequalities for trigonometric polynomials in a symmetric space. Proc. Seventh Winter School “Mathematical Programming and Related Questions”, Drogobych, 1974, Vol. 1: Theory of Functions and Functional Analysis. Moscow: Central. Ekonom.-Mat. Inst. Akad. Nauk SSSR, 1976, pp. 133–140 (in Russian).

12.   Smailov E.S. On the effect of the geometric properties of the spectrum of a polynomial on S. M. Nikol’skii inequalities of different metrics. Siberian Math. J., 1998, vol. 39, no. 5, pp. 1000–1006. doi: 10.1007/BF02672923 

13.   Nursultanov E.D. Nikol’skii inequality for different metrics and properties of the sequence of norms of the fourier sums of a function in the Lorentz space. Proc. Steklov Inst. Math., 2006, vol. 255, pp. 185–202. doi: 10.1134/S0081543806040158

14.   Gogatishvili A., Opic B., Tikhonov S., Trebels W. Ulyanov-type inequalities between Lorentz–Zygmund spaces. J. Fourier Anal. Appl., 2014, vol. 20, no. 5, pp. 1020–1049. doi: 10.1007/s00041-014-9343-4 .

15.   Sherstneva L.A. Nikol’skij inequalities for trigonometric polynomials in Lorentz spaces. Mosc. Univ. Math. Bull., 1984, vol. 39, no. 4, pp. 75–81.

16.   Shvelidze N.V. On imbedding theorems in some functional spaces. Bull. Acad. Sci. Georgian SSR, 1976, vol. 83, no. 2, pp. 290–292 (in Russian).

17.   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).

18.   Sherstneva L.A. On the properties of best Lorentz approximations and certain embedding theorems. Soviet Math. (Iz. VUZ), 1987, vol. 31, no. 10, pp. 62–73.

19.   Sharpley R. Space $\Lambda_{\alpha}(X)$ and interpolation. J. Func. Anal., 1972, vol. 11, no. 4, pp. 479–513. doi: 10.1016/0022-1236(72)90068-7 .

20.   Bennet C., Rudnik K. On Lorentz–Zygmund spaces. Disser. Math., vol. 175, Warszawa, 1979, 66 p. ISBN: 8301011114 .

21.   Edmunds D.E., Evans W.D. Hardy operators, function spaces and embedding. Berlin; Heidelberg: Springer-Verlag, 2004, 328 p. ISBN: 3-540-21972-2/hbk .

22.   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 .

23.   Rodin V.A. The Hardy–Littlewood theorem for the cosine series in a symmetric space. Math. Notes, 1976, vol. 20, no. 2, pp. 693–696. doi: 10.1007/BF01155876 .

24.   Komissarov A.A. O nekotorykh svoistvakh funktsional’nykh sistem [On some properties of functional systems]. Available from VINITI, no. 5827-83, Moscow, 1983, 28 p.

25.   Ditzian Z., Prymak A. Nikol’skii inequalities for Lorentz spaces. Rocky Mountain Jour. Math., 2010, vol. 40, no. 1, pp. 209–223. DOI: 10.1216/RMJ-2010-40-1-209 .

26.   Sabziev N.M. Nekotorye ekstremal’nye svoistva funktsii iz klassa $L_p$ (Some extremal properties of a function from the class $L_p$). Available from VINITI, no. 5252-82, Baku, 1982, 12 p.