Multiple sine series and Nikol’skii classes in uniform metric
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.
KeywordsMultiple sine series Mixed modulus of smoothness Nikol’skii classes Generalized “small” Lipschitz classes
Mathematics Subject Classification42B05 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.
- 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
- 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.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
- 17.Pak, I.N.: Fourier coefficients and the Lipschitz class. Anal. Math. 16(1), 57–64 (1990) (in Russian)Google Scholar
- 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.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.Tevzadze, T.Sh: On certain classes of functions and Fourier series. Trudy Tbiliss. Univ. 149–150, 51–64 (1973) (in Russian)Google Scholar
- 27.Zygmund, A.: Trigonometric Series, vol. 2, 2nd edn. Cambridge University Press, New York (1959)Google Scholar