Recovery of continuous functions from their Fourier coefficients known with error

Abstract

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.