On the approximation of conjugate functions from \(L^{p}\) by some special matrix means of conjugate Fourier series

2014 ◽  
Vol 54 (2) ◽  
Author(s):  
Radosława Krantz ◽  
Aleksandra Rzepka
Keyword(s):  
Fourier Series ◽  
Conjugate Functions ◽  
Conjugate Fourier Series ◽  
Matrix Means ◽  
Special Matrix

scholarly journals On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
W. Łenski ◽  
B. Szal
Keyword(s):  
Fourier Series ◽  
Conjugate Functions ◽  
Pointwise Approximation ◽  
Conjugate Fourier Series ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Norm Approximation

The results generalizing some theorems onN, pnE, γsummability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.

scholarly journals Degrees of the approximations by some special matrix means of conjugate Fourier series

2019 ◽  
Vol 52 (1) ◽  
pp. 116-129
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka ◽  
Ewa Sylwestrzak-Maślanka
Keyword(s):  
Fourier Series ◽  
Conjugate Fourier Series ◽  
Matrix Means ◽  
Special Matrix

Abstract In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

scholarly journals On the Approximation of Functions From Lp by Some Special Matrix Means of Fourier Series

2015 ◽  
Vol 55 (1) ◽  
pp. 81-90
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix ◽  
Norm Approximation ◽  
Appl Math

Abstract The results corresponding to some theorems of S. Lal [Appl. Math. and Comput. 209 (2009), 346-350] and the results of W. Łenski and B. Szal [Banach Center Publ., 95, (2011), 339-351] are shown. The better degrees of pointwise approximation than these in mentioned papers by another assumptions on summability methods for considered functions are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated.

scholarly journals On Pointwise Approximation of Functions by Some Matrix Means of Conjugate Fourier Series

2015 ◽  
Vol 55 (1) ◽  
pp. 91-108
Author(s):  
W. Lenski ◽  
B. Szal
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Conjugate Fourier Series ◽  
Riesz Method ◽  
Matrix Means ◽  
Special Cases ◽  
Norm Approximation

Abstract The results corresponding to some theorems of S. Lal [Tamkang J. Math., 31(4)(2000), 279-288] and the results of the authors [Banach Center Publ. 92(2011), 237-247] are shown. The same degrees of pointwise approximation as in mentioned papers by significantly weaker assumptions on considered functions are obtained. From presented pointwise results the estimation on norm approximation with essentialy better degrees are derived. Some special cases as corollaries for iteration of the Nörlund or the Riesz method with the Euler one are also formulated.

scholarly journals On estimates of deviation of conjugate functions from matrix operators of their Fourier series by some expressions with r-differences of the entries

2021 ◽  
Vol 109 (123) ◽  
pp. 109-123
Author(s):  
Włodzimierz Łenski ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Conjugate Functions ◽  
Conjugate Fourier Series ◽  
Matrix Operators

We extend the results of the authors from [Abstract and Applied Analysis, Volume 2016, Article ID 9712878] to the case conjugate Fourier series.

scholarly journals The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties

2021 ◽  
Vol 57 (2) ◽  
pp. 156-175
Author(s):  
P. G. Patseika ◽  
Y. A. Rouba
Keyword(s):  
Fourier Series ◽  
Asymptotic Expression ◽  
Conjugate Function ◽  
Conjugate Functions ◽  
Chebyshev Series ◽  
Approximation Properties ◽  
Computational Mathematics ◽  
Conjugate Fourier Series ◽  
Optimal Value ◽  
Polynomial Series

Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

scholarly journals Degrees of the approximation of integrable functions by some special matrix means of fourier series

2019 ◽  
Vol 105 (119) ◽  
pp. 39-47
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka ◽  
Ewa Sylwestrzak-Maślanka
Keyword(s):  
Fourier Series ◽  
Integrable Functions ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix ◽  
Approximation Of Function

We consider the pointwise and normwise approximation of function by some special matrix means of its Fourier series. The results corresponding to the theorem of ?enski and Szal in [5] and the results of Saini and Singh in [6] are shown. Some special cases as corollaries are also formulated.

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