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 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 Approximation by double Walsh polynomials

1992 ◽  
Vol 15 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Ferenc Móricz
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Partial Sums ◽  
Lipschitz Classes ◽  
Sufficient Conditions For Convergence ◽  
By Products ◽  
De La Vallée Poussin

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach spaceX. In particular,Xmay beLp(I2), where1≦p<∞andI2=[0,1)×[0,1), orCW(I2), the latter being the collection of uniformlyW-continuous functions onI2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence inLp(I2)-norm and uniform convergence onI2as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.

Relation between Fourier series and Wiener algebras

2021 ◽  
Vol 18 (1) ◽  
pp. 80-103
Author(s):  
Roald Trigub
Keyword(s):  
Fourier Series ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Real Variable ◽  
State University ◽  
Approximation Properties ◽  
Function Classes ◽  
Almost Everywhere ◽  
Necessary And Sufficient ◽  
Complex Valued

New relations between the Banach algebras of absolutely convergent Fourier integrals of complex-valued measures of Wiener and various issues of trigonometric Fourier series (see classical monographs by A.~Zygmund [1] and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new properties of the Fourier series from the known properties of the Wiener algebras, as well as new results to be obtained for those algebras from the known properties of Fourier series. For example, criteria, i.e. simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to be a Fourier series, or the Fourier series of a function of bounded variation, and so forth. Approximation properties of various linear summability methods of Fourier series (comparison, approximation of function classes and single functions) and summability almost everywhere (often with the set indication) are considered. The presented material was reported by the author on 12.02.2021 at the Zoom-seminar on the theory of real variable functions at the Moscow State University.

The maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series

2008 ◽  
Vol 45 (3) ◽  
pp. 321-331
Author(s):  
István Blahota ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Hardy Space ◽  
Maximal Operator ◽  
Fejér Means

In this paper we prove that the maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series is not bounded from the Hardy space H2/3 ( G2 ) to the space L2/3 ( G2 ).

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

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