### Characteristics of Besov-Nikol'skiĭ class of functions.

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We study weighted ${L}_{p}$-integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight ${x}^{-\alpha}$ depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

In this paper we study embedding theorems for function classes which are subclasses of ${L}_{p}$, 1 ≤ p ≤ ∞. To define these classes, we use the notion of best trigonometric approximation as well as that of a (λ,β)-derivative, which is the generalization of a fractional derivative. Estimates of best approximations of transformed Fourier series are obtained.

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