Recovery of continuous functions from their Fourier coefficients known with error

K.V. Pozharska (Institute of Mathematics of NAS of Ukraine),
O.A. Pozharskyi (Institute of Mathematics of NAS of Ukraine)


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.


Fourier series; methods of regularization; $$$\Lambda$$$-methods of summation

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