The problem of optimal recovery is considered for functions from their Fourier coefficients known with error. In a more general statement,this problem for the classes of smooth and atalytic functions defined on various compact manifolds can be found in the classical paper byG.G. Magaril-Il'yaev, K.Y. Osipenko.Namely, the paper is devoted to the recovery of continuous real-valued functions $y$ of one variable from the classes $W^{\psi}_{p}$, $1 \leq p< \infty$,that are defined in terms of generalized smoothness $\psi$ from their Fourier coefficients with respect to some complete orthonormal in the space $L_2$ system$\Phi = \{ \varphi_k \}_{k=1}^{\infty}$ of continuous functions, that are blurred by noise.Assume that for function $y$ we know the values $y_k^{\delta}$ of their noisy Fourier coefficients, besides $y_k^{\delta} = y_k + \delta \xi_k$, $k = 1,2, \dots$,where $y_k$ are the corresponding Fourier coefficients, $\delta \in (0,1)$, and $\xi = (\xi_k)_{k=1}^{\infty}$ is a noise.Additionally let the functions from the system $\Phi$ be continuous and satisfy the condition $\| \varphi_k \|_{C}\leq C_1 k^{\beta}$, $k=1,2,\dots$,where $C_1>0$, $\beta \geq 0$ are some constants, and $\| \cdot\|_{C}$ is the standart norm of the space $C$ of continuous on the segment $[0,1]$ functions.Under certain conditions on parameter $\psi$, we obtain order estimates of the approximation errors of functions from the classes$$W^{\psi}_{p} = \left\{ y \in L_2\colon  \| y \|^p_{W^{\psi}_{p}} = \sum\limits_{k=1}^{\infty} \psi^p(k) |y_k|^p \leq 1 \right\}, \quad 1 \leq p< \infty,$$in metric of the space $C$ by the so-called$\Lambda$-method of series summation that is defined by the number triangular matrix $\Lambda = \{ \lambda_k^n \}_{k=1}^n$, $n=n(\delta) \in \mathbb{N}$,with some restrictions on its elements.Note, that we extend the known results [8, 7] to a more wide spectrum of the classesof functions and for a more general restrictions on the noise level.In our results a case is considered when the noise is stronger than those in the space $l_2$ of real number sequences, but not stochastic.

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into L 2 (D, ̺ D ) multiplied with the supremum of the Christoffel function of the subspace spanned by the first m singular functions. Here the measure ̺ D is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a √ log n and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

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