Startseite On moduli of smoothness and averaged differences of fractional order
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On moduli of smoothness and averaged differences of fractional order

  • Yurii Kolomoitsev EMAIL logo
Veröffentlicht/Copyright: 8. August 2017
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 20 Heft 4

Abstract

We consider two types of fractional integral moduli of smoothness, which are widely used in the theory of functions and approximation theory. In particular, we obtain new equivalences between these moduli of smoothness and the classical moduli of smoothness. It turns out that for the fractional integral moduli of smoothness some pathological effects arise.

MSC 2010: Primary 26A33; Secondary 47B39; 33E30; 42A45; 42A10
Key Words and Phrases: modulus of smoothness; averaged differences; fractional order; Fourier multipliers

Acknowledgements

The author thanks to R.M. Trigub for the proposed problem and to A.A. Dovgoshey for the valuable discussions.

This research has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 704030.

References

[1] R. Akgün, Realization and characterization of modulus of smoothness in weighted Lebesgue spaces. St. Petersburg Math. J. 26, No 5 (2015), 741–756.10.1090/spmj/1356Suche in Google Scholar

[2] P.L. Butzer, H. Dyckhoff, E. Görlich, R.L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes. Can. J. Math. 29 (1977), 781–793.10.4153/CJM-1977-081-6Suche in Google Scholar

[3] R.A. DeVore, G.G. Lorentz, Constructive Approximation. Springer-Verlag, New York (1993).10.1007/978-3-662-02888-9Suche in Google Scholar

[4] Z. Ditzian, V. Hristov, K. Ivanov, Moduli of smoothness and K-functional in Lp, 0 < p < 1. Constr. Approx. 11 (1995), 67–83.10.1007/BF01294339Suche in Google Scholar

[5] V.I. Ivanov, Direct and inverse theorems of approximation theory in the metrics Lp for 0 < p < 1. Math. Notes18, No 5 (1975), 972–982.10.1007/BF01153563Suche in Google Scholar

[6] V. Kokilashvili, S. Samko, A refined inverse inequality of approximation in weighted variable exponent Lebesgue spaces. Proc. A. Razmadze Math. Inst. 151 (2009), 134–138.Suche in Google Scholar

[7] Yu. Kolomoitsev, The inequality of Nikolskii-Stechkin-Boas type with fractional derivatives in Lp for 0 < p < 1. (In Russian). Trudy Inst. Prikl. Mat. Mekh. 15 (2007), 115–119.Suche in Google Scholar

[8] Yu. Kolomoitsev, On moduli of smoothness and K-functionals of fractional order in the Hardy spaces. J. Math. Sci. 181, No 1 (2012), 78–97; Transl. from: Ukr. Mat. Visn. 8, No 3 (2011), 421–446.10.1007/s10958-012-0677-7Suche in Google Scholar

[9] Yu.S. Kolomoitsev, Approximation properties of generalized Bochner-Riesz means in the Hardy spaces Hp, 0 < p ≤ 1 (In Russian). Mat. Sb. 203, No 8 (2012), 79–96; Transl. in: Sb. Math. 203, No 7-8 (2012), 1151–1168.Suche in Google Scholar

[10] Yu.S. Kolomoitsev, R.M. Trigub, On the nonclassical approximation method for periodic functions by trigonometric polynomials. J. Math. Sci. (N. Y.) 188, No 2 (2013), 113–127.10.1007/s10958-012-1111-xSuche in Google Scholar

[11] E. Liflyand, S. Samko, R. Trigub, The Wiener algebra of absolutely convergent Fourier integrals: An overviev. Anal. Math. Phys. 2, No 1 (2012), 1–68.10.1007/s13324-012-0025-6Suche in Google Scholar

[12] N.N. Pustovoitov, An estimate for the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences, and the multidimensional Jackson theorem (In Russian). Mat. Sb. 188, No 10 (1997), 95–108; Transl. in: Sb. Math. 188, No 10 (1997), 1507–1520.10.1070/SM1997v188n10ABEH000265Suche in Google Scholar

[13] K.V. Runovskii, On approximation by families of linear polynomial operators in the spaces Lp, 0 <p < 1 (In Russian). Mat. Sb. 185, No 8 (1994), 81–102; Transl. in: Russian Acad. Sci. Sb. Math. 82, No 2 (1995), 441–459.Suche in Google Scholar

[14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Science Publishers, New York (1993).Suche in Google Scholar

[15] S.B. Stechkin, On the order of the best approximations of continuous functions (In Russian). Izv. Akad. Nauk SSSR Ser. Mat. 15, No 3 (1951), 219–242.Suche in Google Scholar

[16] E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971).10.1515/9781400883899Suche in Google Scholar

[17] E.A. Storozhenko, V.G. Krotov, P. Oswald, Direct and converse theorems of Jackson type in Lp spaces, 0 <p < 1. Math. USSR-Sb. 27 (1975), 355–374.10.1070/SM1975v027n03ABEH002519Suche in Google Scholar

[18] E.A. Storozhenko, P. Oswald, Jackson’s theorem in the spaces Lp(ℝk), 0 <p < 1 (In Russian). Sibirsk. Mat. Z. 19, No 4 (1978), 888–901.Suche in Google Scholar

[19] R. Taberski, Differences, moduli and derivatives of fractional orders. Commentat. Math. 19 (1976-77), 389–400.Suche in Google Scholar

[20] A.F. Timan, Theory of Approximation of Functions of a Real Variable. Internat. Ser. of Monographs on Pure and Applied Mathematics, Vol. 34, Pergamon Press, Oxford etc. (1963).10.1016/B978-0-08-009929-3.50008-7Suche in Google Scholar

[21] R.M. Trigub, Summability of Fourier series and certain questions of approximation theory (In Russian). Deposited in: VINITI, 5145–80 (1980).Suche in Google Scholar

[22] R.M. Trigub, Fourier multipliers and K-functionals in spaces of smooth functions (In Russian). Ukr. Mat. Visn. 2, No 2 (2005), 236–280; Transl. in: Ukr. Math. Bull. 2, No 2 (2005), 239–284.Suche in Google Scholar

[23] R.M. Trigub, The exact order of approximation of periodic functions by Bernstein-Stechkin polynomials (In Russian). Mat. Sb. 204, No 12 (2013), 127–146; Transl. in: Sb. Math. 204, No 11-12 (2013), 1819–1838.Suche in Google Scholar

[24] R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions. Kluwer - Springer (2004).10.1007/978-1-4020-2876-2Suche in Google Scholar

Received: 2016-11-22
Revised: 2017-6-22
Published Online: 2017-8-8
Published in Print: 2017-8-28

© 2017 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
  5. Perturbation methods for nonlocal Kirchhoff–type problems
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  26. Corrigendum
  27. Corrigendum: Fractional integral on martingale hardy spaces with variable exponents
https://doi.org/10.1515/fca-2017-0051
Schlagwörter für diesen Artikel
modulus of smoothness; averaged differences; fractional order; Fourier multipliers

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
  5. Perturbation methods for nonlocal Kirchhoff–type problems
  6. Research Paper
  7. On infinite order differential operators in fractional viscoelasticity
  8. Research Paper
  9. Fundamental solution of the multi-dimensional time fractional telegraph equation
  10. Research Paper
  11. Robustness and convergence of fractional systems and their applications to adaptive schemes
  12. Research Paper
  13. Stability analysis of linear distributed order fractional systems with distributed delays
  14. Research Paper
  15. Fractional sobolev spaces and functions of bounded variation of one variable
  16. Research Paper
  17. Approximate controllability for fractional differential equations of sobolev type via properties on resolvent operators
  18. Research Paper
  19. On moduli of smoothness and averaged differences of fractional order
  20. Research Paper
  21. A piecewise memory principle for fractional derivatives
  22. Research Paper
  23. A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels
  24. Shopt Paper
  25. Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method
  26. Corrigendum
  27. Corrigendum: Fractional integral on martingale hardy spaces with variable exponents
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