Approximation of functions from Lp(\widetildeω)βby linear operators of conjugate Fourier series

2011 ◽  
Vol 92 ◽  
pp. 237-247 ◽  
Author(s):  
WŁodzimierz Łenski ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
Approximation Of Functions ◽  
Conjugate Fourier Series

scholarly journals Some direct and inverse theorems in approximation of functions

1983 ◽  
Vol 34 (2) ◽  
pp. 143-154 ◽  
Author(s):  
R. N. Mohapatra ◽  
D. C. Russell
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
Approximation Of Functions ◽  
Inverse Results ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems ◽  
Degree Of Convergence

AbstractThe paper is concerned with the determination of the degree of convergence of a sequence of linear operators connected with the Fourier series of a function of class Lp (p > 1) to that function and some inverse results in relating the convergence to the classes of functions. In certain cases one can obtain the saturation results too. In all cases Lp norm is used.

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.

ON APPROXIMATION OF FUNCTIONS FROM ZYGMUND CLASSES BY BIHARMONIC POISSON INTEGRALS

2021 ◽  
Vol 4 ◽  
Author(s):  
Bogdan Borsuk ◽  
◽  
Alexander Khanin ◽  
Keyword(s):  
Fourier Series ◽  
Applied Mathematics ◽  
Linear Operators ◽  
Systematic Research ◽  
Elliptic Type ◽  
Approximation Of Functions ◽  
Linear Positive Operators ◽  
Poisson Integrals ◽  
Direct Use ◽  
Poisson Type

The paper is devoted to a behavior investigation of the upper bound of deviation of functions from Zygmund classes from their biharmonic Poisson integrals. Systematic research in this direction was conducted by a number of Ukrainian as well as foreign scientists. But most of the known results relate to an estimation of deviations of functions from different classes from operators that were constructed based on triangular l-methods of the Fourier series summation (Fejer, Valle Poussin, Riesz, Rogozinsky, Steklov, Favard, etc.). Concerning the results relating to linear methods of the Fourier series summation, given by a set of functions of natural argument (Abel-Poisson, Gauss-Weierstrass, biharmonic and threeharmonic Poisson integrals), in this direction the progress was less notable. This may be due to the fact that the above-mentioned linear methods the Fourier series summation are solutions of corresponding integral and differential equations of elliptic type. And, therefore, they require more time-consuming calculations in order to obtain some estimates, that are suitable for a direct use for applied purposes. At the same time, in the present paper we investigate approximative characteristics of linear positive Poisson-type operators on Zygmund classes of functions. According to the well-known results by P.P. Korovkin, these positive linear operators realize the best asymptotic approximation of functions from Zygmund classes. Thus, the estimate obtained in this paper for the deviation of functions from Zygmund classes from their biharmonic Poisson integrals (the least studied and most valuable among all linear positive operators) is relevant from the viewpoint of applied mathematics.

scholarly journals Pointwise Approximation of Functions from by Linear Operators of Their Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Włodzimierz Łenski ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Lipschitz Classes ◽  
Matrix Summability

We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise approximation of functions from the pointwise integral Lipschitz classes by matrix summability means of their Fourier series as well as the theorems on norm approximations.

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