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 Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

2019 ◽  
pp. 18-34
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Chebyshev Series ◽  
Approximation Properties ◽  
Fejér Means ◽  
Markov System ◽  
Asymptotic Expressions ◽  
Uniform Approximations ◽  
Optimal Value

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

scholarly journals On approximations of the function |x|s by the Vallee Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions

2019 ◽  
Vol 55 (3) ◽  
pp. 263-282
Author(s):  
P. G. Patseika ◽  
Y. A. Rovba
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Asymptotic Expression ◽  
Uniform Estimate ◽  
Main Section ◽  
Computational Mathematics ◽  
Power Singularity ◽  
Asymptotic Estimation ◽  
Optimal Value ◽  
Pointwise Estimation

The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

scholarly journals On approximation in generalized Zygmund class

2019 ◽  
Vol 52 (1) ◽  
pp. 370-387
Author(s):  
Hare Krishna Nigam
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Conjugate Function ◽  
Degree Of Approximation ◽  
Zygmund Class ◽  
Conjugate Fourier Series ◽  
Product Method ◽  
Hausdorff Means

AbstractHere, we estimate the degree of approximation of a conjugate function {\tilde g} and a derived conjugate function {\tilde g'} , of a 2π-periodic function g \in Z_r^\lambda , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced from our main theorems. We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

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 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.

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