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 $C^{\alpha,r}_{\beta,p}$, 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 $C^{\alpha,r}_{\beta,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1\leq p\leq \infty$, with generalized Poisson kernels $$ P_{\alpha,r,\beta}(t)=\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos \big(kt-\frac{\beta\pi}{2}\big), \ \alpha>0, r>0, \ \beta\in \mathbb{R}.$$