Abstract
In variable exponent Lebesgue spaces, the equivalence between the modulus of smoothness given by one-sided Steklov means and realization functionals that use Zygmund–Riesz and Euler means is established. A class of functions equivalent to generalized moduli of smoothness of the order r ∈ \(\mathbb{N}\) is described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
R. A. DeVore and G. G. Lorentz, Constructive Approximation (Springer, Berlin, 1993).
Z. Ditzian, V. H. Hristov, and K. G. Ivanov, “Moduli of smoothness and K-functionals in L p, 0 < p < 1,” Constr. Approx. 11 (1), 67–83 (1995). https://doi.org/10.1007/BF01294339
R. Akgün, “Realization and characterization of modulus of smoothness in weighted Lebesgue spaces,” Algebra Anal. 26 (5), 64–87 (2014);
St. Petersburg Math. J. 26 (5), 741–756 (2015). https://doi.org/10.1090/spmj/1356
A. F. Timan, The Theory of Approximation of Functions of a Real Variable (Fizmatgiz, Moscow, 1960; Pergamon Press, Oxford, 1963).
S. Z. Jafarov, “Derivatives of a polynomial of the best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent,” Demonstr. Math. 50 (1), 245–251 (2017). https://doi.org/10.1515/dema-2017-0023
A. Testici, “On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Leebesgue space with variable exponent,” Period. Math. Hungar. 80 (1), 59–73 (2020).
S. Tikhonov, “On moduli of smoothness of fractional order,” Real Anal. Exchange 30 (2), 507–518 (2004/2005).
W. Orlicz, “Über conjugierte Exponentenfolgen,” Studia Math. 3, 200–211 (1931).
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034 (Springer, Berlin, 1983). https://doi.org/10.1007/BFb0072210
A. N. Kolmogorov, “Zur normierbarkeit eines allgemeinen topologischen linearen räumen,” Studia Math. 5 (1), 29–38 (1934).
I. I. Sharapudinov, “Topology of the space \({{\mathcal{L}}^{{p(t)}}}\)([0, 1]),” Math. Notes Acad. Sci. SSSR (4), 796–806 (1979). https://doi.org/10.1007/BF01159546
O. Kováčik and J. Rákosník, “On spaces L p(x) and W k, p(x),” Czechoslovak Math. J. 41 (4), 592–618 (1991).
L. Diening, P. Harjulehto, P. Hästö, and M. Rúžička. Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017 (Springer, Berlin, 2011). https://doi.org/10.1007/978-3-642-18363-8
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Springer, Basel, 2013).
I. I. Sharapudinov, “Uniform boundedness in L p (p = p(x)) of some families of convolution operators,” Math. Notes 59 (2), 205–212 (1996). https://doi.org/10.1007/BF02310962
D. M. Israfilov, V. Kokilashvili, S. Samko, “Approximation in weighted Lebesgue and Smirnov spaces with variable exponent,” Proc. A. Razmadze Math. Inst. 143, 45–55 (2007).
A. Guven and D. M. Israfilov, “Trigonometric approximation in generalized Lebesgue space L p(x),” J. Math. Inequal. 4 (2), 285–299 (2010). https://doi.org/10.7153/jmi-04-25
R. Akgün, “Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent,” Ukr. Math. J. 63 (1), 1–26 (2011). https://doi.org/10.1007/s11253-011-0485-0
R. Akgün and V. Kokilashvili, “On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces,” Banach J. Math. Anal. 5 (1), 70–82 (2011). https://doi.org/10.15352/bjma/1313362981
R. Akgün and V. Kokilashvili, “Some notes on trigonometric approximation of (α, ψ)-differentiable functions in weighted variable exponent Lebesgue spaces,” Proc. A. Razmadze Math. Inst. 163, 15–23 (2013).
I. I. Sharapudinov, “Approximation of functions in \(L_{{2\pi }}^{{p(x)}}\) by trigonometric polynomials,” Izv. Math. 77 (2), 407–434 (2013). https://doi.org/10.1070/IM2013v077n02ABEH00264110.1070/IM2013v077n02ABEH002641
S. S. Volosivets, “Approximation of functions and their conjugates in variable Lebesgue spaces,” Sb. Math. 208 (1), 44–59 (2017). https://doi.org/10.1070/SM8636
N. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian].
D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable L p spaces,” Ann. Acad. Sci. Fenn. 31, 239–264 (2006).
I. I. Sharapudinov, “Approximation of smooth functions in \(L_{{2\pi }}^{{p(x)}}\) by Vallee-Poussin means,” Izv. Saratov. Univ. Ser. Mat. Mekh. Inf. 13 (1 (1)), 45–49 (2013). https://doi.org/10.18500/1816-9791-2013-13-1-1-45-49
S. S. Volosivets and A. A. Tyuleneva, “Approximation of functions and their conjugates in L p and uniform metric by Euler means,” Demonstr. Math. 51 (1), 141–150 (2018). https://doi.org/10.1515/dema-2018-0013
S. S. Volosivets, “Modified modulus of smoothness and approximation in weighted Lorentz spaces by Borel and Euler means,” Probl. Anal. Issues Anal. 10 (28) (1), 87–100 (2021). https://doi.org/10.15393/j3.art.2021.8950
M. F. Timan, “Best approximations of a function and linear methods of summation of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat. 29 (3), 587–604 (1965).
R. M. Trigub, “The constructive characteristics of certain classes of functions,” Izv. Akad. Nauk SSSR, Ser. Mat. 29 (3), 615–630 (1965).
S. B. Stechkin, “On absolute convergence of Fourier series. II,” Izv. Akad. Nauk SSSR, Ser. Mat. 19 (4), 221–246 (1955).
Funding
This study is supported by the Ministry of Education of the Russian Federation within fulfilling state task, project no. FSRR-2020-0006.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by M. Talacheva
About this article
Cite this article
Volosivets, S.S. Realization Functionals and Description of a Modulus of Smoothness in Variable Exponent Lebesgue Spaces. Russ Math. 66, 8–19 (2022). https://doi.org/10.3103/S1066369X22060081
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X22060081