Approximation of functions by linear summation methods in the Orlicz-type spaces

Chaichenko, Stanislav; Savchuk, V. V.; Shidlich, Andrii

doi:10.1007/s10958-020-04967-y

Cited by 4 publications

(1 citation statement)

References 14 publications

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“…Stepanets and others. Among recent results in this direction, we refer to the following papers: [9], where order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions were obtained, [1] for direct and inverse theorems of approximation by the methods of Zygmund, Abel-Poisson, Taylor-Abel-Poisson in the Orlicz-type spaces, and [11] for the investigation of saturation of several linear methods of summation of the Fourier series.…”

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confidence: 99%

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

Pozharska

Pozharskyi

2021

Carpathian Math. Publ.

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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 \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]

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Section: Problem Statement and History Overviewmentioning

confidence: 99%

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

Pozharska

Pozharskyi

2021

Carpathian Math. Publ.

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Approximation by linear means of Vilenkin–Fourier series in Stepanets spaces

Mingachev,

Volosivets

2024

European Journal of Mathematics

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Approximation of classes of Poisson integrals by rectangular Fejér means

Rovenska

2024

Front. Appl. Math. Stat.

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The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane.

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Approximation of functions by linear summation methods in the Orlicz-type spaces

Cited by 4 publications

References 14 publications

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

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

Approximation by linear means of Vilenkin–Fourier series in Stepanets spaces

Approximation of classes of Poisson integrals by rectangular Fejér means

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