### Recovery of continuous functions from their Fourier coefficients known with error

#### Abstract

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.

#### Keywords

#### Full Text:

PDF#### References

Abdullayev F., Chaichenko S., Imash Kyzy M., Shidlich A. "Direct and inverse approximation theorems in the weighted Orlicz-type spaces with a variable exponent", *Turkish J. Math.*, 2020; 44(1): pp. 284-299. doi:10.3906/mat-1911-3

Babenko V.F., Gun'ko M.S. "On the optimal recovery of convolution of n-functions by linear information", *Ukrainian Math. J.*, 2016; 68(5): pp. 579-585. (in Russian) doi:10.1007/s11253-016-1248-8

Gun'ko M.S., Babenko V.F., Parfinovych N.V. "Optimal recovery of elements from Hilbert space and their scalar products by Fourier coefficients known with error", *Ukrainian Math. J.*, 2020; 72(6): pp. 736-750. doi:10.37863/umzh.v72i6.1107

Il'in V.A., Poznyak È.G. *Foundations of mathematical analysis. Part II. A course in higher mathematics and mathematical physics. 4th ed.*, Moscow, Fizmatlit, 2002. (in Russian)

Korneichuk N.P. *Exact constants in approximation theory*, Nauka, Moscow, 1987; 424 p. (in Russian)

Magaril-Il'yaev G.G., Osipenko K.Y. "Optimal recovery of functions and their derivatives from Fourier coefficients prescribed with an error", *SB MATH*, 2002; 193(3): pp. 387-407. doi:10.1070/SM2002v193n03ABEH000637

Mathe P., Pereverzev S.V. "Stable summation of orthogonal series with noisy coefficients", *J. Approx. Theory*, 2002; 118: pp. 66-80. doi:10.1006/jath.2002.3710

Sharipov K. "On the recovery of continuous functions from noisy Fourier coefficients", *Comput. Methods in Appl. Math.*, 2011; 11(1): pp. 75-82. doi:10.2478/cmam-2011-0004

Stepanets A.I. *Classification and approximation of periodic functions*, Kyiv, Nauk. Dumka, 1987. (in Russian)

Stepanets A.I. "Best approximations of $$$q$$$-ellipsoids in spaces $$$S^{p, \mu}_{\varphi}$$$", *Ukrainian Math. J.*, 2004; 56: pp. 1646-1652. doi:10.1007/s11253-005-0140-8

Stepanets A.I., Rukasov V.I. "Spaces $$$S^p$$$ with nonsymmetric metric", *Ukrainian Math. J.*, 2003; 55: pp. 322-338. doi:10.1023/A:1025472514408

Tikhonov A.N. "On stable summability methods for Fourier series", *Dokl. Akad. Nauk SSSR*, 1964; 156(2): 268-271. (in Russian)

Tikhonov A.N. "On the solution of ill-posed problems and the method of regularization", *Dokl. Akad. Nauk SSSR*, 1963; 151(3): 501-504. (in Russian)

Tikhonov A.N., Arsenin V.Ja. *Methods for the solution of ill-posed problems. 2nd ed.*, Moscow, Nauka, 1979. (in Russian)

DOI: https://doi.org/10.15421/242008

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