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Approximation of integrable functions by general linear matrix operators of their Fourier series

Mishra Vishnu Narayan, Łenski Włodzimierz, Szal Bogdan

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

136--152

EN

The pointwise estimates of the deviation Tn,Af(⋅)−f(⋅) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of f , with application of the rth differences of the entries of A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.

2

Pointwise convergence of Fourier-Laguerre series of integrable functions

Łenski W., Szal B.

Fasciculi Mathematici

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2018

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Nr 60

93--101

EN

We extend and improve the some results of Xh. Z. Krasniqi [Int. J. of Anal. and Appl. Vol. 1, 33-39 (2013)], M. L. Mittal and M. V. Singh [Operators, Int. J. of Analysis, Vol. 2015, Article ID 478345, 4 pages] and from many other papers on summability of Fourier-Laguerre series to strong summability proving the estimate of the deviation of the partial sums from considered functions. There also is a remark on summability methods used in cited papers.

3

Point wise approximation of 2π/r-periodic functions by matrix operators of their Fourier series with r-differences of the entries

Łenski W., Szal B.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

309--322

EN

We extend the results of Xh. Z. Krasniqi [Acta Comment. Univ. Tartu. Math., 2013, 17, 89-101] and the authors [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] to the case of 2π/r-periodic functions. More over, as a measure of approximation r-differences of the entries are used.

4

Approximation by q-analogue of modified Jakimovski-Leviatan-Stancu type operators

Mursaleen M., Nasiruzzaman M., Wafi A.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

175--189

EN

In this paper, we introduce the q-analogue of the Jakimovski-Leviatan type modied operators introduced by Atakut with the help of the q-Appell polynomials. We obtain some approximation results via the well-known Korovkin’s theorem for these operators. We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover, we study the rate of convergence in terms of modulus of continuity of these operators in a weighted space.

5

Approximation of conjugate functions by general linear operators of their Fourier series at the Lebesgue points

Łenski W., Szal B.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

878--892

EN

The pointwise estimates of the deviations (…) and (…) in terms of moduli of continuity (…) and (…) are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal [3, Theorem 1, p. 437].

6

Approximation of functions from IP {uj)p by linear operators of their Fourier series

Łenski W., Szal B.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 2

189-202

EN

We show the results corresponding to theorems of S. Lai [Appl. Math. Comput., 209 (2009) 346-350] on the rate of approximation of functions from the generalized integral Lipschitz classes by matrix summability means of their Fourier series as well as to the authors theorems [Acta Comment. Univ. Tartu. Math., 13 (2009), 11-24] also on such approximations.