Based on the Steklov operator, we consider a modulus of smoothness for functions in some Banach function spaces, which can be not translation invariant, and establish its main properties. A constructive characterization of the Lipschitz class is obtained with the help of the Jackson-type direct theorem and the inverse theorem on trigonometric approximation. As an application, we present several examples of related (weighted) function spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Abilov and F. V. Abilova, “Some problems of the approximation of 2π -periodic functions by Fourier sums in the space\({L}_{2\pi }^{2}\) ,” Math. Notes, 76, No. 5-6, 749–757 (2004).
R. Akgün, “Polynomial approximation in weighted Lebesgue spaces,” East J. Approx., 17, No. 3, 253–266 (2011).
R. Akgün, “Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent,” Ukr. Math. Zh., 63, No. 1, 3–23 (2011); English translation: Ukr. Math. J., 63, No. 1, 1–26 (2011).
R. Akgün, “Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth,” Georgian Math. J., 18, No. 2, 203–235 (2011).
R. Akgün, “Exponential approximation in variable exponent Lebesgue spaces on the real line,” Constr. Math. Anal., 5, No. 4, 214–237 (2022).
R. Akg¨un and D. M. Israfilov, “Approximation in weighted Orlicz spaces,” Math. Slovaca, 61, No. 4, 601–618 (2011).
V. V. Arestov, “Integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR Ser. Mat., 45, No. 1, 3–22 (1981).
S. N. Bernstein, Collected works, Vol. 1: Constructive Theory of Functions [1905–1930], Izd. Akad. Nauk SSSR, Moscow (1952).
C. Bennett and R. Sharpley, “Interpolation of operators,” Pure Appl. Math., 129, Academic Press, Boston, MA (1988).
S. Bloom and R. Kerman, “Weighted LΦ integral inequalities for operators of Hardy type,” Studia Math., 110, No. 1, 35–52 (1994).
D. Cruz-Uribe, L. Diening, and P. Hästö, “The maximal operator on weighted variable Lebesgue spaces,” Fract. Calc. Appl. Anal., 14, No. 3, 361–374 (2011).
F. Dai, “Jackson-type inequality for doubling weights on the sphere,” Constr. Approx., 24, No. 1, 91–112 (2006).
R. A. De Vore and G. G. Lorentz, “Constructive approximation,” Fundamental Principles of Mathematical Sciences, 303, Springer-Verlag, Berlin (1993).
L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, “Lebesgue and Sobolev spaces with variable exponents,” Lecture Notes Math., 2017, Springer-Verlag, Heidelberg (2011).
Z. Ditzian, “Rearrangement invariance and relations among measures of smoothness,” Acta Math. Hungar., 135, No. 3, 270–285 (2012).
Z. Ditzian and K. G. Ivanov, “Strong converse inequalities,” J. Anal. Math., 61, 61–111 (1993).
E. A. Gadjieva, Investigation of the Properties of Functions with Quasimonotone Fourier Coefficients in Generalized Nikolskii–Besov Spaces [in Russian], Author’s Abstract of the PhD Thesis, Tbilisi (1986).
A. Guven and D. M. Israfilov, “Approximation by trigonometric polynomials in weighted rearrangement invariant spaces,” Glas. Mat., Ser. III, 44 (64), No. 2, 423–446 (2009).
D. M. Israfilov and A. Guven, “Approximation by trigonometric polynomials in weighted Orlicz spaces,” Studia Math., 174, No. 2, 147–168 (2006).
S. Z. Jafarov, “Approximation by Fejér sums of Fourier trigonometric series in weighted Orlicz space,” Hacet. J. Math. Stat., 43, No. 3, 259–268 (2013).
S. Z. Jafarov, “On moduli of smoothness in Orlicz classes,” Proc. Inst. Math. Mech. Azerb. Natl. Acad. Sci. Azerb., 33, No. 51, 85–92 (2010).
A. Y. Karlovich, “Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces,” J. Integral Equat. Appl., 15, No. 3, 263–320 (2003).
M. Khabazi, “The mean convergence of trigonometric Fourier series in weighted Orlicz classes,” Proc. Razmadze Math. Inst., 129, 65–75 (2002).
O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Math. J., 41 (116), No. 4, 592–618 (1991).
I. I. Sharapudinov, “On the uniform boundedness in Lp (p = p(x)) of some families of convolution operators,” Math. Notes, 59, No. 1–2, 205–212 (1996).
I. I. Sharapudinov, “Some problems in approximation theory in the spaces Lp(x)(E),” Anal. Math., 33, No. 2, 135–153 (2007).
I. I. Sharapudinov, “Approximation of functions in \({L}_{2\pi }^{p\left(\bullet \right)}\) by trigonometric polynomials,” Izv. Ros. Akad. Nauk, Ser. Mat., 77, No. 2, 197–224 (2013).
R. Taberski, “Approximation of functions possessing derivatives of positive orders,” Ann. Polon. Math., 34, No. 1, 13–23 (1977).
9A. F. Timan, “Theory of approximation of functions of a real variable,” Internat. Ser. Monogr., Pure Appl. Math., 34, Pergamon Press Book, Macmillan Co., New York (1963).
A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, New York (1959).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 8, pp. 1015–1031, August, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i8.970.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Akgün, R. A Modulus of Smoothness for Some Banach Function Spaces. Ukr Math J 75, 1159–1177 (2024). https://doi.org/10.1007/s11253-023-02253-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-023-02253-z