Revista Matemática Complutense

, Volume 27, Issue 1, pp 69–91 | Cite as

Functions of bounded second \(p\)-variation

  • Sorina Barza
  • Pilar Silvestre


The generalized functionals of Merentes type generate a scale of spaces connecting the class of functions of bounded second \(p\)-variation with the Sobolev space of functions with p-integrable second derivative. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of order \(2-1/p\). These results extend the results in Lind (Math Inequal Appl 16:2139, 2013) for functions of bounded variation but are not consequence of the last.


Partition Periodic function Bounded (second) Modulus of \(p\)-continuity Bounded second (\({p, \alpha }\)) variation  Fractional moduli of continuity Steklov averages 

Mathematics Subject Classification (2000)

Primary 26A45 Secondary 46E35 



The authors are very grateful to Prof. Viktor Kolyada and their friend Martin Lind for encouragement and discussions, and for pointing out the reference [11]. The first named author would like to acknowledge the travel grant from SVeFUM (2012) which made possible the visit to the University of Barcelona when a part of this research was done. The second named author gratefully acknowledges the opportunity to work as a postdoc in Aalto University with Prof. J. Kinnunen where a part of this project was done. We would like to thank the referees for their remarks, which have improved the final version of the paper.


  1. 1.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  2. 2.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  3. 3.
    Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals, I. Math. Z. 27, 565–606 (1928)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Lind, M.: On fractional smoothness of functions related to \(p\)-variation. Math. Inequal. Appl. 16(1), 2139 (2013)Google Scholar
  5. 5.
    Merentes, N.: On functions of bounded (p,2)-variation. Collect. Math. 43(2), 117–123 (1992)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar Publishing Co., New York (1955)Google Scholar
  7. 7.
    Riesz, F.: Untersuchungen über Systeme integrierbarer Funktionen. Matematishe Annalen 69, 449–497 (1910)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Terehin, A.P.: Approximation of functions of bounded \(p\)-variations. Izv. Vyssh. Uchebn. Zaved. Mat. 2, 171–187 (1965) (Russian)Google Scholar
  9. 9.
    Terehin, A.P.: Functions of bounded \(p\)-variations with a given modulus of continuity. Mat. Zametki 53(6), 523–530 (1972) (Russian); English transl. in. Math. Notes 12, 751–755 (1972)Google Scholar
  10. 10.
    de la Valée Poussin, Ch. J.: Sur la convergence des formules d’interpolation entre ordennées equidistantes. Bull. Acad. Sci. Belg. 314–410 (1908)Google Scholar
  11. 11.
    Volosivets, S.S.: Asymptotic properties of one compact set of smooth functions in the space of functions of bounded \(p\)-variation. Math. Notes 57(1–2), 148–157 (1995)Google Scholar
  12. 12.
    Volosivets, S.S.: The best approximation polynomials and relation between continuity modules in spaces of functions of a bounded \(p\)-variation. Russ. Math. (Iz.VUZ) 40(9), 21–26 (1996)Google Scholar
  13. 13.
    Wiener, N.: The quadratic variation of a function and its Fourier coefficients. J. Math. Phys. 3, 72–94 (1924)zbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceKarlstad UniversityKarlstadSweden
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland

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