Banach Journal of Mathematical Analysis

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

Ramazan Akgun and Vakhtang Kokilashvili

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362981

Digital Object Identifier
doi:10.15352/bjma/1313362981

Mathematical Reviews number (MathSciNet)
MR2738521

Zentralblatt MATH identifier
1206.42002

Subjects
Primary: 42A10: Trigonometric approximation
Secondary: 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol s kii-type inequalities) 26A33: Fractional derivatives and integrals 41A20: Approximation by rational functions 41A25: Rate of convergence, degree of approximation 41A27: Inverse theorems

Keywords
weighted fractional moduli of smoothness converse theorem fractional derivative

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


Export citation

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.