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 by
G.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 classes
of 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.