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.

Keywords

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

Mathematics Subject Classification

42B05 42B35 42A32 

Notes

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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Copyright information

© 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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