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Multiple sine series and Nikol’skii classes in uniform metric

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Abstract

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.

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References

  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)

    MathSciNet  Google Scholar 

  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)

  3. Boas Jr., R.P.: Fourier series with positive coefficients. J. Math. Anal. Appl. 17, 463–483 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Donskikh, S.L.: Multiple Fourier series for functions from a Zygmund class. Moscow Univ. Math. Bull. 65(1), 1–9 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dyachenko, M.I.: Trigonometric series with generalized-monotone coefficients. Izv. Vyssh. Uchebn. Zaved. Mat. 1986(7), 39–50 (1986) (in Russian)

  7. Dyachenko, M.I., Tikhonov, S.Yu.: Smoothness and asymptotic properties of functions with general monotone Fourier coefficients. J. Fourier Anal. Appl. https://doi.org/10.1007/s00041-017-9553-7

  8. Fülöp, V.: Double cosine series with nonnegative coefficients. Acta Sci. Math. (Szeged) 70(1–2), 91–100 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Fülöp, V.: Double sine series with nonnegative coefficients and Lipschitz classes. Colloq. Math. 105(1), 25–34 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fülöp, V., Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Zygmund classes of functions. Analysis (Munich) 28(3), 345–354 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Liflyand, E., Tikhonov, S., Zeltser, M.: Extending tests for convergence of number series. J. Math. Anal. Appl. 377(1), 194–206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lorentz, G.G.: Fourier-Koeffizienten und Funktionenklassen. Math. Z. 51(2), 135–149 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  14. Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Lipschitz classes of functions. Acta Math. Hungar. 121(1–2), 1–19 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Németh, J.: Fourier series with positive coefficients and generalized Lipschitz classes. Acta Sci. Math. (Szeged) 54(3–4), 291–304 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Németh, J.: Note on Fourier series with nonnegative coefficients. Acta Sci. Math. (Szeged) 55(1–2), 83–93 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Pak, I.N.: Fourier coefficients and the Lipschitz class. Anal. Math. 16(1), 57–64 (1990) (in Russian)

  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)

  19. Rubinstein, A.I.: On \(\omega \)-lacunary series and functions from classes \(H^\omega \). Mat. Sb. 65(107), 239–271 (1964) (in Russian)

  20. Tevzadze, T.Sh: On certain classes of functions and Fourier series. Trudy Tbiliss. Univ. 149–150, 51–64 (1973) (in Russian)

  21. Tikhonov, S.: On generalized Lipschitz classes and Fourier series. Z. Anal. Anwend. 23(4), 745–764 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tikhonov, S.: Smoothness conditions and Fourier series. Math. Inequal. Appl. 10(2), 229–242 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Tikhonov, S.: Trigonometric series of Nikol’skii classes. Acta Math. Hungar. 114(1–2), 61–78 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tikhonov, S.: Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153(1), 19–39 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Volosivets, S.S.: Multiple Fourier coefficients and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 427(2), 1070–1083 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yu, D.: Double trigonometric series with positive coefficients. Anal. Math. 35(2), 149–167 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zygmund, A.: Trigonometric Series, vol. 2, 2nd edn. Cambridge University Press, New York (1959)

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Acknowledgements

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

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  1. Department of Mechanics and Mathematics, Saratov State University, 83 Astrakhanskaya St., 410012, Saratov, Russia

    Sergey Volosivets

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  1. Sergey Volosivets
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Correspondence to Sergey Volosivets.

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Volosivets, S. Multiple sine series and Nikol’skii classes in uniform metric. European Journal of Mathematics 5, 206–222 (2019). https://doi.org/10.1007/s40879-018-0226-0

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