Classification of periodic functions and the rate of convergence of their Fourier series

The author proposes a classification for periodic functions that is based on grouping them according to the rate at which their Fourier coefficients tend to zero. The classes $L_\beta^\psi\mathfrak N$ thereby introduced coincide, for fixed values of the defining parameters, with the known classes $W^r$, $W^rH_\omega$, $W_\beta^r$, $W_\beta^rH_\omega$, and the like. Such an approach permits the classification of a wide spectrum of periodic functions, including infinitely differentiable, analytic, and entire functions. The asymptotic behavior of the deviations of the Fourier sums in these classes is studied. The assertions obtained in this direction contain results known earlier on approximation by Fourier sums of classes of differentiable functions.
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