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 $W^{\psi}_{p}$, $1 \leq p0$, $\beta \geq 0$ 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.

Keywords

• fourier series
• methods of regularization
• $\lambda$-methods of summation