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.

Recovery of continuous functions of two variables from their Fourier coefficients known with error

“…As to "ill-posed" problems (for the perturbed Fourier coefficients of respective functions), for a more detailed information concerning the univariate case we refer to the paper [5].…”

“…Let us estimate first the term I 1 from (5). Taking into account the condition (4) and norm properties, we obtain…”

“…It remains to estimate the term I 5 from the right-hand side of (5). Note, that the condition (4) yields uniform boundness of the elements λ n i,j of the matrix Λ, moreover |λ n i,j | ≤ C 2 + 1, i, j = 1, n, n ∈ N. So, we can write…”

“…Remark 2. The estimate of the quantity ∆(W ψ 1,p , Λ, l p ), 1 ≤ p < ∞, for the introduced above classes W ψ 1,p of univariate functions is obtained in the paper [5] (with appropriately modified restrictions (4) on the elements λ n k , k = 1, . .…”