On Approximation of Functions in the Generalized Zygmund Class Using $$(E,r)(N,q_n)$$ Mean Associated with Conjugate Fourier Series

2021 ◽  
pp. 235-248
Author(s):  
B. P. Padhy ◽  
Susanta Kumar Paikray ◽  
Anwesha Mishra ◽  
U. K. Misra
Keyword(s):  
Fourier Series ◽  
Approximation Of Functions ◽  
Zygmund Class ◽  
Conjugate Fourier Series

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.

Approximation of Signals in the Weighted Zygmund Class Via Euler-hausdorff Product Summability Mean of Fourier Series

2020 ◽  
Vol 87 (1-2) ◽  
pp. 22
Author(s):  
A. A. Das ◽  
S. K. Paikray ◽  
T Pradhan ◽  
H. Dutta
Keyword(s):  
Fourier Series ◽  
Order Of Convergence ◽  
Approximation Of Functions ◽  
Zygmund Class ◽  
Approximation Of Signals

Approximation of functions of Lipschitz and zygmund classes have been considered by various researchers under different summability means. In the proposed paper, we have studied an estimation of the order of convergence of Fourier series in the weighted Zygmund class <em>W(Z<sub>r</sub><sup>(ω)</sup>)</em> by using Euler-Hausdorff product summability mean and accordingly established some (presumably new) results. Moreover, the results obtained here are the generalization of several known results.

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 Best approximation of conjugate of a function in generalized Zygmund

2019 ◽  
Vol 50 (4) ◽  
pp. 417-427
Author(s):  
Hare Krishna Nigam
Keyword(s):  
Fourier Series ◽  
Error Estimates ◽  
Best Approximation ◽  
Degree Of Approximation ◽  
Time Study ◽  
Zygmund Class ◽  
Conjugate Fourier Series ◽  
First Time

In this paper, we, for the very first time, study the error estimates of conjugate of a function ~g of g(2-periodic) in generalized Zygmund class Y wz (z 1); by Matix-Euler (TEq) product operatorof conjugate Fourier series. In fact, we establish two theorems on degree of approximation of afunction ~g of g (2-periodic) in generalized Zygmund class Y wz (z 1); by Matix-Euler (TEq)product means of its conjugate Fourier series. Our main theorem generalizes three previouslyknown results. Thus the results of [7], [8] and [26] become the particular cases of our Theorem2.1. Some corollaries are also deduced from our main theorem.

scholarly journals Absolute and strong summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series and $r$ times differentiated conjugate Fourier series by matrix methods

2021 ◽  
pp. 74
Author(s):  
N.T. Polovina
Keyword(s):  
Fourier Series ◽  
Strong Summability ◽  
Matrix Methods ◽  
Conjugate Fourier Series ◽  
The Matrix

We establish conditions of $|\gamma|_p$- and $[\gamma]_p$-summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series at the point where $\gamma = \| \gamma_{nk} \|$ is the matrix of transformation of series to sequence. Analogous conditions are considered also for $r$ times differentiated conjugate Fourier series.

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