Approximation by Fourier sums on the classes of generalized Poisson integrals

Abstract

We present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C_{\beta ,p}^{\alpha ,r}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2\pi$--periodic functions $C_{\beta ,p}^{\alpha ,r}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_p$, $1\le p\le \infty$, with generalized Poisson kernels $$P_{\alpha ,r,\beta }(t)=\sum _{k=1}^{\infty }{e}^{-\alpha k^r}\cos (kt-\frac{\beta \pi }{2}),\alpha >0,r>0,\beta \in \mathbb{R}.$$

Survey article