Recovery of continuous functions of two variables from their Fourier coefficients known with error
In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensi...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2021-12-01
|
| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5124 |
| _version_ | 1827281265209901056 |
|---|---|
| author | K.V. Pozharska A.A. Pozharskyi |
| author_facet | K.V. Pozharska A.A. Pozharskyi |
| author_sort | K.V. Pozharska |
| collection | DOAJ |
| description | In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions.
As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions.
We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\] |
| first_indexed | 2024-04-24T08:56:36Z |
| format | Article |
| id | doaj.art-30b199e91591480ea479eb5bfeb98d20 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:56:36Z |
| publishDate | 2021-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-30b199e91591480ea479eb5bfeb98d202024-04-16T07:09:07ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-12-0113367668610.15330/cmp.13.3.676-6864451Recovery of continuous functions of two variables from their Fourier coefficients known with errorK.V. Pozharska0https://orcid.org/0000-0001-7599-8117A.A. Pozharskyi1Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, UkraineInstitute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, UkraineIn this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions. As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions. We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]https://journals.pnu.edu.ua/index.php/cmp/article/view/5124fourier seriesmethod of regularization$\lambda$-method of summation |
| spellingShingle | K.V. Pozharska A.A. Pozharskyi Recovery of continuous functions of two variables from their Fourier coefficients known with error Karpatsʹkì Matematičnì Publìkacìï fourier series method of regularization $\lambda$-method of summation |
| title | Recovery of continuous functions of two variables from their Fourier coefficients known with error |
| title_full | Recovery of continuous functions of two variables from their Fourier coefficients known with error |
| title_fullStr | Recovery of continuous functions of two variables from their Fourier coefficients known with error |
| title_full_unstemmed | Recovery of continuous functions of two variables from their Fourier coefficients known with error |
| title_short | Recovery of continuous functions of two variables from their Fourier coefficients known with error |
| title_sort | recovery of continuous functions of two variables from their fourier coefficients known with error |
| topic | fourier series method of regularization $\lambda$-method of summation |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5124 |
| work_keys_str_mv | AT kvpozharska recoveryofcontinuousfunctionsoftwovariablesfromtheirfouriercoefficientsknownwitherror AT aapozharskyi recoveryofcontinuousfunctionsoftwovariablesfromtheirfouriercoefficientsknownwitherror |