Multiple sine series and Nikol’skii classes in uniform metric

  • Sergey Volosivets
Research Article


We give necessary and sufficient conditions for a function odd in each variable to belong to Nikol’skii classes defined via mixed modulus of smoothness and mixed derivative (both have arbitrary integer orders). These conditions are given in terms of growth of partial sums of Fourier sine coefficients with power weights or rate of decreasing to zero of these coefficients. A similar problem for generalized “small” Lipschitz classes is also treated.


Multiple sine series Mixed modulus of smoothness Nikol’skii classes Generalized “small” Lipschitz classes 

Mathematics Subject Classification

42B05 42B35 42A32 



The author thanks both referees for their critical comments and valuable suggestions which helped to improve the results of paper and its presentation.


  1. 1.
    Antonov, A.P.: Classes \({\rm Lip}(\alpha, p)\) for double trigonometric series with monotone coefficients. Moscow Univ. Math. Bull. 63(1), 12–16 (2010)MathSciNetGoogle Scholar
  2. 2.
    Bary, N.K., Stechkin, S.B.: Best approximations and differential properties of two conjugate functions. Tr. Mosk. Mat. Obs. 5, 483–522 (1956) (in Russian)Google Scholar
  3. 3.
    Boas Jr., R.P.: Fourier series with positive coefficients. J. Math. Anal. Appl. 17, 463–483 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Butzer, P.L., Dyckhoff, H., Görlich, E., Stens, R.L.: Best trigonometric approximations, fractional order derivatives and Lipschitz classes. Canad. J. Math. 29(4), 781–793 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Donskikh, S.L.: Multiple Fourier series for functions from a Zygmund class. Moscow Univ. Math. Bull. 65(1), 1–9 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dyachenko, M.I.: Trigonometric series with generalized-monotone coefficients. Izv. Vyssh. Uchebn. Zaved. Mat. 1986(7), 39–50 (1986) (in Russian)Google Scholar
  7. 7.
    Dyachenko, M.I., Tikhonov, S.Yu.: Smoothness and asymptotic properties of functions with general monotone Fourier coefficients. J. Fourier Anal. Appl.
  8. 8.
    Fülöp, V.: Double cosine series with nonnegative coefficients. Acta Sci. Math. (Szeged) 70(1–2), 91–100 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fülöp, V.: Double sine series with nonnegative coefficients and Lipschitz classes. Colloq. Math. 105(1), 25–34 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fülöp, V., Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Zygmund classes of functions. Analysis (Munich) 28(3), 345–354 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Han, D., Li, G., Yu, D.: Double sine series and higher order Lipschitz classes of functions. Bull. Math. Anal. Appl. 5(1), 10–21 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Liflyand, E., Tikhonov, S., Zeltser, M.: Extending tests for convergence of number series. J. Math. Anal. Appl. 377(1), 194–206 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lorentz, G.G.: Fourier-Koeffizienten und Funktionenklassen. Math. Z. 51(2), 135–149 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Lipschitz classes of functions. Acta Math. Hungar. 121(1–2), 1–19 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Németh, J.: Fourier series with positive coefficients and generalized Lipschitz classes. Acta Sci. Math. (Szeged) 54(3–4), 291–304 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Németh, J.: Note on Fourier series with nonnegative coefficients. Acta Sci. Math. (Szeged) 55(1–2), 83–93 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pak, I.N.: Fourier coefficients and the Lipschitz class. Anal. Math. 16(1), 57–64 (1990) (in Russian)Google Scholar
  18. 18.
    Potapov, M.K., Simonov, B.V., Tikhonov, S.Yu.: Mixed moduli of smoothness in \(L_p, 1<p<\infty \): a survey. Surv. Approx. Theory 8, 1–57 (2013)Google Scholar
  19. 19.
    Rubinstein, A.I.: On \(\omega \)-lacunary series and functions from classes \(H^\omega \). Mat. Sb. 65(107), 239–271 (1964) (in Russian)Google Scholar
  20. 20.
    Tevzadze, T.Sh: On certain classes of functions and Fourier series. Trudy Tbiliss. Univ. 149–150, 51–64 (1973) (in Russian)Google Scholar
  21. 21.
    Tikhonov, S.: On generalized Lipschitz classes and Fourier series. Z. Anal. Anwend. 23(4), 745–764 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tikhonov, S.: Smoothness conditions and Fourier series. Math. Inequal. Appl. 10(2), 229–242 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Tikhonov, S.: Trigonometric series of Nikol’skii classes. Acta Math. Hungar. 114(1–2), 61–78 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tikhonov, S.: Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153(1), 19–39 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Volosivets, S.S.: Multiple Fourier coefficients and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 427(2), 1070–1083 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yu, D.: Double trigonometric series with positive coefficients. Anal. Math. 35(2), 149–167 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zygmund, A.: Trigonometric Series, vol. 2, 2nd edn. Cambridge University Press, New York (1959)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsSaratov State UniversitySaratovRussia

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