## Banach Journal of Mathematical Analysis

### On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces

#### Abstract

In this work we prove improved converse theorems of trigonometric approximation in variable exponent Lebesgue spaces with some Muckenhoupt weights.

#### Article information

Source
Banach J. Math. Anal., Volume 5, Number 1 (2011), 70-82.

Dates
First available in Project Euclid: 14 August 2011

https://projecteuclid.org/euclid.bjma/1313362981

Digital Object Identifier
doi:10.15352/bjma/1313362981

Mathematical Reviews number (MathSciNet)
MR2738521

Zentralblatt MATH identifier
1206.42002

#### Citation

Akgun, Ramazan; Kokilashvili , Vakhtang. On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces. Banach J. Math. Anal. 5 (2011), no. 1, 70--82. doi:10.15352/bjma/1313362981. https://projecteuclid.org/euclid.bjma/1313362981

#### References

• R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz spaces, J. Funct. Spaces Appl. (to appear).
• –-, Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 152 (2010), 1–18.
• –-, Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth, Georgian Math. J. (to appear).
• R. Akgün and D.M. Israfilov, Approximation and moduli of fractional orders in Smirnov-Orlicz classes, Glas. Mat. Ser. III 43(63) (2008), no. 1, 121–136.
• R. Akgün and V. Kokilashvili, The refined direct and converse inequalities of trigonometric approximation in weighted variable exponent Lebesgue spaces, Georgian Math. J. (to appear).
• E.A. Hadjieva, Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolskii-Besov spaces, Author's summary of dissertation, Tbilisi, 1986, (Russian).
• P. Hästö and L. Diening, Muckenhoupt weights in variable exponent spaces, Preprint, Albert Ludwings Universität Freiburg, Mathematische Fakultät, http://www.helsinki.fi/~ pharjule/varsob/publications.shtml.
• D.M. Israfilov, V. Kokilashvili and S.G. Samko, Approximation in weighted Lebesgue spaces and Smirnov spaces with variable exponents, Proc. A. Razmadze Math. Inst. 143 (2007) 25–35.
• V. Kokilashvili, The converse theorem of constructive theory of functions in Orlicz spaces, Soobshch. Akad. Nauk Gruzin. SSR 37 (1965), No. 2, 263–270 (Russian).
• –-, On approximation of periodic functions, Soobshch Akad. Nauk. GruzSSR 34 (1968), 51–81. (Russian)
• V. Kokilashvili and S.G. Samko, Singular integrals weighted Lebesgue spaces with variable exponent, Georgian M. J. 10 (2003), No:1, 145–156.
• –-, Operators of Harmonis Analysis in weighted spaces with non-standard growth, J. Math. Anal. Appl. 352 (2009), 15–34.
• –-, A refined inverse inequality of approximation in weighted variable exponent Lebesgue spaces, Proc. A. Razmadze Math. Inst. 151 (2009), 134–138.
• V. Kokilashvili and Y.E. Yildirir, On the approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 143 (2007), 103–113.
• –-, The Estimation of High Order Generalized Modulus of Cotinuity in $L_\omega ^p$, Proc. A. Razmadze Math. Inst. 143% (2007), 135–137.
• N. Korneĭchuk, Exact constants in approximation theory, Encyclopedia of Mathematics and its Applications, 38, Cambridge University Press, Cambridge, 1991.
• P.P. Petrushev and V.A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne and Sydney, 1987.
• Bl. Sendov and V.A. Popov, The averaged moduli of smoothness with applications in numerical methods and approximation, John Wiley & Sons, New York, 1988.
• S.B. Stechkin, On the order of the best approximation of continuous functions, Izv. Akad. Nauk. SSSR, Ser. Mat. 15 (1951), 219–242.
• A.F. Timan, Theory of approximation of functions of a real variable, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian).
• A.F. Timan and M.F. Timan, The generalized modulus of continuity and best mean approximation(Russian), Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 17–20.
• M.F. Timan, Inverse theorems of the constructive theory of functions in the spaces $L_p$, Mat. Sb. 46(88) (1958), 125–132. (Russian)
• A. Zygmund, A remark on the integral modulus of continuity, Univ. Nac. Tucumán. Revista A. 7 (1950), 259–269.