Skip to main content
SpringerLink
Log in

Inequalities in Approximation Theory Involving Fractional Smoothness in L p, 0 < p < 1

  • Chapter
  • First Online:
Topics in Classical and Modern Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in L p, 0 < p < 1. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness ω β(f (α), δ)p, where f (α) is a fractional derivative of the function f, are derived. A description of the class of functions with the optimal rate of decrease of a fractional modulus of smoothness is given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. E. Belinsky, E. Liflyand, Approximation properties in L p, 0 < p < 1. Funct. Approx. Comment. Math. 22, 189–199 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Y.A. Brudnyi, Criteria for the existence of derivatives in L p. Math. USSR-Sb. 2(1), 35–55 (1967)

    Article  Google Scholar 

  3. Y.A. Brudnyi, On the saturation class of a spline approximation with uniform nodes in L p, 0 < p < ∞. Issled. Teor. Funkts. Mnogikh Veshchestv. Perem. 34–40 (1984) (in Russian)

    Google Scholar 

  4. P.L. Butzer, U. Westphal, An access to fractional differentiation via fractional difference quotients, in Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol. 457 (Springer, Berlin, 1975), pp. 116–145

    Google Scholar 

  5. P.L. Butzer, H. Dyckhoff, E. Görlich, R.L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes. Can. J. Math. 29, 781–793 (1977)

    Article  MathSciNet  Google Scholar 

  6. P. Civin, Inequalities for trigonometric integrals. Duke Math. J. 8, 656–665 (1941)

    Article  MathSciNet  Google Scholar 

  7. J. Czipszer, G. Freud, Sur l’approximation d’une fonction périodique et de ses dérivées successives par un polynome trigonométrique et par ses dérivées successives. Acta Math. 99, 33–51 (1958)

    Article  MathSciNet  Google Scholar 

  8. R.A. DeVore, G.G. Lorentz, Constructive Approximation (Springer, New York, 1993)

    Book  Google Scholar 

  9. Z. Ditzian, A note on simultaneous polynomial approximation in L p[−1,  1], 0 < p < 1. J. Approx. Theory 82(2), 317–319 (1995)

    Article  MathSciNet  Google Scholar 

  10. Z. Ditzian, S. Tikhonov, Ul’yanov and Nikol’skii-type inequalities. J. Approx. Theory 133(1), 100–133 (2005)

    Article  MathSciNet  Google Scholar 

  11. Z. Ditzian, S. Tikhonov, Moduli of smoothness of functions and their derivatives. Stud. Math. 180(2), 143–160 (2007)

    Article  MathSciNet  Google Scholar 

  12. Z. Ditzian, V. Hristov, K. Ivanov, Moduli of smoothness and K-functional in L p, 0 < p < 1. Constr. Approx. 11, 67–83 (1995)

    Article  MathSciNet  Google Scholar 

  13. G.H. Hardy, J. Littlewood, Some properties of fractional integrals. Math. Z. 27, 565–606 (1928)

    Article  MathSciNet  Google Scholar 

  14. V.I. Ivanov, Direct and inverse theorems of approximation theory in the metrics L p for 0 < p < 1. Math. Notes 18(5), 972–982 (1975)

    Article  Google Scholar 

  15. H. Johnen, K. Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, in Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach 1976). Lecture Notes in Mathematics, vol. 571 (Springer, Berlin, 1977), pp. 119–140

    Google Scholar 

  16. Y.S. Kolomoitsev, Description of a class of functions with the condition ω r(f, h)p ≤ Mh r−1+1∕p for 0 < p < 1. Vestn. Dnepr. Univ. Ser. Mat. 8, 31–43 (2003) (in Russian)

    Google Scholar 

  17. Y. Kolomoitsev, The inequality of Nikol’skii-Stechkin-Boas type with fractional derivatives in L p, 0 < p < 1. Tr. Inst. Prikl. Mat. Mekh. 15, 115–119 (2007) (in Russian)

    MATH  Google Scholar 

  18. Y. Kolomoitsev, On moduli of smoothness and K-functionals of fractional order in the Hardy spaces. J. Math. Sci. 181(1), 78–97 (2012); translation from Ukr. Mat. Visn. 8(3), 421–446 (2011)

    Google Scholar 

  19. Y. Kolomoitsev, On a class of functions representable as a Fourier integral. Tr. Inst. Prikl. Mat. Mekh. 25, 125–132 (2012) (in Russian)

    MathSciNet  MATH  Google Scholar 

  20. Y. Kolomoitsev, Multiplicative sufficient conditions for Fourier multipliers. Izv. Math. 78(2), 354–374 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78(2), 145–166 (2014)

    Google Scholar 

  21. Y. Kolomoitsev, Best approximations and moduli of smoothness of functions and their derivatives in L p, 0 < p < 1. J. Approx. Theory 232, 12–42 (2018)

    Article  MathSciNet  Google Scholar 

  22. Y. Kolomoitsev, J. Prestin, Sharp estimates of approximation of periodic functions in Hölder spaces. J. Approx. Theory 200, 68–91 (2015)

    Article  MathSciNet  Google Scholar 

  23. K.A. Kopotun, On K-monotone polynomial and spline approximation in L p, 0 < p < ∞ (quasi)norm, in Approximation Theory VIII, ed. by C. Chui, L. Schumaker (World Scientific, Singapore, 1995), pp. 295–302

    MATH  Google Scholar 

  24. K.A. Kopotun, On equivalence of moduli of smoothness of splines in L p, 0 < p < 1. J. Approx. Theory 143(1), 36–43 (2006)

    Article  MathSciNet  Google Scholar 

  25. V.G. Krotov, On differentiability of functions in L p, 0 < p < 1. Sb. Math. USSR 25 , 101–119 (1983)

    Article  Google Scholar 

  26. S.M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, New York, 1975)

    Book  Google Scholar 

  27. M.K. Potapov, B.V. Simonov, S.Y. Tikhonov, Fractional Moduli of Smoothness (Max Press, Moscow, 2016)

    Google Scholar 

  28. J. Peetre, A remark on Sobolev spaces. The case 0 < p < 1. J. Approx. Theory 13, 218–228 (1975)

    Google Scholar 

  29. P.P. Petrushev, V.A. Popov, Rational Approximation of Real Functions (Cambridge University Press, Cambridge, 1987)

    MATH  Google Scholar 

  30. T.V. Radoslavova, Decrease orders of the L p-moduli of continuity (0 < p ≤∞). Anal. Math. 5(3), 219–234 (1979)

    Article  MathSciNet  Google Scholar 

  31. K. Runovski, Approximation of families of linear polynomial operators. Dissertation of Doctor of Science, Moscow State University, 2010

    MATH  Google Scholar 

  32. K. Runovski, H.-J. Schmeisser, General moduli of smoothness and approximation by families of linear polynomial operators, in New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis (Birkhäuser, Cham, 2014), pp. 269–298

    Google Scholar 

  33. K.V. Runovskii, Approximation by trigonometric polynomials, K-functionals and generalized moduli of smoothness. Sb. Math. 208(1–2), 237–254 (2017)

    Article  MathSciNet  Google Scholar 

  34. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, in Theory and Applications (Gordon and Breach, Yverdon, 1993)

    MATH  Google Scholar 

  35. B.V. Simonov, S.Y. Tikhonov, Embedding theorems in constructive approximation. Sb. Math. 199(9), 1367–1407 (2008); translation from Mat. Sb. 199(9), 107–148 (2008)

    Google Scholar 

  36. B. Simonov, S. Tikhonov, Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theory 162(9), 1654–1684 (2010)

    Article  MathSciNet  Google Scholar 

  37. E.A. Storozhenko, P. Oswald, Jackson’s theorem in the spaces L p(ℝk), 0 < p < 1. Sib. Math. J. 19(4), 630–656 (1978)

    Article  Google Scholar 

  38. E.A. Storozhenko, V.G. Krotov, P. Oswald, Direct and inverse theorems of Jackson type in the space L p, 0 < p < 1. Mat. Sb. 98(3), 395–415 (1975)

    MathSciNet  Google Scholar 

  39. R. Taberski, Differences, moduli and derivatives of fractional orders. Commentat. Math. 19, 389–400 (1976–1977)

    MathSciNet  MATH  Google Scholar 

  40. R. Taberski, Approximation in the Fréchet spaces L p (0 < p ≤ 1). Funct. Approx. Comment. Math. 7, 105–121 (1979)

    MathSciNet  MATH  Google Scholar 

  41. R. Taberski, Aproksymacja funkcji wielomianami trygonometrycznymi. Wydawnictwo Naukowe UAM, Poznan, 1979 (in Polish)

    Google Scholar 

  42. R. Taberski, Trigonometric approximation in the norms and seminorms. Stud. Math. 80, 197–217 (1984)

    Article  MathSciNet  Google Scholar 

  43. S. Tikhonov, On modulus of smoothness of fractional order. Real Anal. Exchange 30, 507–518 (2004/2005)

    Article  Google Scholar 

  44. M.F. Timan, Converse theorems of the constructive theory of functions in the spaces L p (1 ≤ p ≤∞). Mat. Sb. 46(88), 125–132 (1958)

    MathSciNet  Google Scholar 

  45. A.F. Timan, Theory of Approximation of Functions of a Real Variable (Pergamon Press, Oxford, 1963)

    MATH  Google Scholar 

  46. R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer, Dordrecht, 2004)

    Book  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität zu Lübeck, Lübeck, Germany

    Yurii Kolomoitsev & Tetiana Lomako

  2. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Slov’yans’k, Donetsk Region, Ukraine

    Yurii Kolomoitsev & Tetiana Lomako

Authors
  1. Yurii Kolomoitsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tetiana Lomako
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yurii Kolomoitsev .

Editor information

Editors and Affiliations

  1. Georgia Southern University, Statesboro, GA, USA

    Martha Abell

  2. Georgia Southern University, Statesboro, GA, USA

    Emil Iacob

  3. Georgia Southern University, Statesboro, GA, USA

    Alex Stokolos

  4. Georgia Southern University, Statesboro, GA, USA

    Sharon Taylor

  5. ICREA, Passeig de Lluís Companys, Barcelona, Spain ; Centre de Recerca Matemàtica, Barcelona, Spain

    Sergey Tikhonov

  6. Georgia Southern University, Statesboro, GA, USA

    Jiehua Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kolomoitsev, Y., Lomako, T. (2019). Inequalities in Approximation Theory Involving Fractional Smoothness in L p, 0 < p < 1. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_12

Download citation

Publish with us

Policies and ethics

Search

Navigation