Recovery of continuous functions from their Fourier coefficients known with error

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'ya...
Ausführliche Beschreibung

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. Ausführliche Beschreibung

Autor*in:	K.V. Pozharska [verfasserIn] O.A. Pozharskyi [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch ; Ukrainisch

Erschienen:	2020

Schlagwörter:	fourier series methods of regularization $\lambda$-methods of summation

Übergeordnetes Werk:	In: Researches in Mathematics - Oles Honchar Dnipro National University, 2019, 28(2020), 2, Seite 24-34
Übergeordnetes Werk:	volume:28 ; year:2020 ; number:2 ; pages:24-34

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DOI / URN:	10.15421/242008

Katalog-ID:	DOAJ055084680