A modulus of smoothness for some Banach function spaces

Ramazan Akgün

doi:10.3842/umzh.v75i8.970

Ramazan Akgün Department of Mathematics, Faculty of Arts and Sciences, Balikesir University, Turkey https://orcid.org/0000-0001-6247-8518

DOI: https://doi.org/10.3842/umzh.v75i8.970

Анотація

УДК 517.5

Модуль гладкості для деяких банахових функціональних просторів

На основі оператора Стєклова розглянуто модуль гладкості функцій у деяких банахових функціональних просторах, які можуть не бути трансляційно-інваріантними, та вивчено його основні властивості. Конструктивну характеристику класу Ліпшиця отримано за допомогою прямої теореми типу Джексона та оберненої теореми про тригонометричне наближення. Як застосування, наведено кілька прикладів відповідних (вагових) функціональних просторів.

Посилання

V. A. Abilov, F. V. Abilova, Some problems of the approximation of $2π$-periodic functions by Fourier sums in the space $L_{2π}^{2}$, Math. Notes, 76, № 5–6, 749–757 (2004). DOI: https://doi.org/10.1023/B:MATN.0000049674.45111.71

R. Akgün, Polynomial approximation in weighted Lebesgue spaces, East J. Approx., 17, № 3, 253–266 (2011).

R. Akgün, Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent, Ukr. Math. J., 63, № 1, 1–26 (2011). DOI: https://doi.org/10.1007/s11253-011-0485-0

R. Akgün, Polynomial approximation of functions in weightrd Lebesgue and Smirnov spaces with nonstandard growth, Georgian. Math. J., 18, № 2, 203–235 (2011). DOI: https://doi.org/10.1515/gmj.2011.0022

R. Akgün, Exponential approximation in variable exponent Lebesgue spaces on the real line, Constr. Math. Anal., 5, № 4, 214–237 (2022). DOI: https://doi.org/10.33205/cma.1167459

R. Akgün, D. M. Israfilov, Approximation in weighted Orlicz spaces, Math. Slovaca, 61, № 4, 601–618 (2011). DOI: https://doi.org/10.2478/s12175-011-0031-4

V. V. Arestov, Integral inequalities for trigonometric polynomials and their derivatives} (in Russian), Izv. Akad. Nauk SSSR Ser. Mat., 45, № 1, 3–22 (1981).

S. N. Bernu{s}tein, Collected works, vol. I, The constructive theory of functions [1905–1930], Izdat. Akad. Nauk SSSR, Moscow (1952).

C. Bennett, R. Sharpley, Interpolation of operators, Pure and Appl. Math., 129, Academic Press, Inc., Boston, MA (1988).

S. Bloom, R. Kerman, Weighted $L_{Phi}$ integral inequalities for operators of Hardy type, Studia Math., 110, № 1, 35–52 (1994). DOI: https://doi.org/10.4064/sm-110-1-35-52

D. Cruz-Uribe, L. Diening, P. Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. and Appl. Anal., 14, № 3, 361–374 (2011). DOI: https://doi.org/10.2478/s13540-011-0023-7

F. Dai, Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 24, № 1, 91–112 (2006). DOI: https://doi.org/10.1007/s00365-005-0614-9

R. A. De Vore, G. G. Lorentz, Constructive approximation, Fundamental Principles of Mathematical Sciences, 303, Springer-Verlag, Berlin (1993).

L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., 2017, Springer-Verlag, Heidelberg (2011). DOI: https://doi.org/10.1007/978-3-642-18363-8

Z. Ditzian, Rearrangement invariance and relations among measures of smoothness, Acta Math. Hungar., 135, № 3, 270–285 (2012). DOI: https://doi.org/10.1007/s10474-011-0171-6

Z. Ditzian, K. G. Ivanov, Strong converse inequalities, J. Anal. Math., 61, 61–111 (1993). DOI: https://doi.org/10.1007/BF02788839

E. A. Gadjieva, Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolskii–Besov spaces} (in Russian), Author's summary of PhD dissertation, Tbilisi (1986).

A. Guven, D. M. Israfilov, Approximation by trigonometric polynomials in weighted rearrangement invariant spaces, Glas. Mat. Ser. III, 44 (64), № 2, 423–446 (2009). DOI: https://doi.org/10.3336/gm.44.2.10

D. M. Israfilov, A. Guven, Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Math., 174, № 2, 147–168 (2006). DOI: https://doi.org/10.4064/sm174-2-3

S. Z. Jafarov, Approximation by Fejér sums of Fourier trigonometric series in weighted Orlicz space, Hacet. J. Math. and Stat., 43, № 3, 259–268 (2013).

S. Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc. Inst. Math. and Mech. Azerb. Natl. Acad. Sci., 33, № 51, 85–92 (2010).

A. Y. Karlovich, Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces, J. Integral Equat. and Appl., 15, № 3, 263–320 (2003). DOI: https://doi.org/10.1216/jiea/1181074970

M. Khabazi, The mean convergence of trigonometric Fourier series in weighted Orlicz classes, Proc. A. Razmadze Math. Inst., 129, 65–75 (2002).

O. Kováčik, J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (116), № 4, 592–618 (1991). DOI: https://doi.org/10.21136/CMJ.1991.102493

I. I. Sharapudinov, On the uniform boundedness in $L^{p}$ $(p=p(x))$ of some families of convolution operators, Math. Notes, 59, № 1–2, 205–212 (1996). DOI: https://doi.org/10.1007/BF02310962

I. I. Sharapudinov, Some problems in approximation theory in the spaces $L^{p(x)}(E)$ (in Russian), Anal. Math., 33, № 2, 135–153 (2007).

I. I. Sharapudinov, Approximation of functions in $L_{2π}^{p(·)}$ by trigonometric polynomials, Izv. Ross. Acad. Nauk Ser. Mat., 77, № 2, 197–224 (2013). DOI: https://doi.org/10.1070/IM2013v077n02ABEH002641

R. Taberski, Approximation of functions possessing derivatives of positive orders, Ann. Polon. Math., 34, № 1, 13–23 (1977). DOI: https://doi.org/10.4064/ap-34-1-13-23

A. F. Timan, Theory of approximation of functions of a real variable, Internat. Ser. Monogr. Pure and App. Math., 34, Pergamon Press Book, The Macmillan Co., New York (1963). DOI: https://doi.org/10.1016/B978-0-08-009929-3.50008-7

A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, New York (1959).