Modulus of Continuity and Best Approximation with Respect to Vilenkin-Like Systems in Some Function Spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 315-332
Author(s):  
István Mező
Keyword(s):  
Fourier Series ◽  
Modulus Of Continuity ◽  
Lebesgue Space ◽  
Best Approximation ◽  
Acta Math ◽  
Function Spaces ◽  
Vilenkin Group ◽  
Strong Connection ◽  
The Best Approximation

Abstract We rephrase Fridli's result [Fridli, Acta Math. Hungar. 45: 393–396, 1985] on the modulus of continuity with respect to a Vilenkin group in the Lebesgue space. We show that this result is valid in the logarithm space and for Vilenkin-like systems. In addition, we prove that there is a strong connection between the best approximation of Fourier series and the modulus of continuity, not only in the Lebesgue space [Gát, Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 17: 161–169, 2001] but in the logarithm space too. We formulate two variable generalizations of the obtained results, which have not been known till now even in the Walsh case.

scholarly journals On approximation by trigonometrical polynomials in $L_2$ space

2016 ◽  
Vol 24 ◽  
pp. 89
Author(s):  
O.V. Polyakov
Keyword(s):  
Modulus Of Continuity ◽  
Best Approximation ◽  
Generalized Modulus Of Continuity ◽  
Jackson Type ◽  
The Best Approximation ◽  
Differentiable Functions

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

scholarly journals Generalization of the Hardy-Littlewood theorem on Fourier series

2021 ◽  
Vol 104 (4) ◽  
pp. 49-55
Author(s):  
S. Bitimkhan ◽  
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Lebesgue Space ◽  
Fourier Coefficients ◽  
Double Fourier Series ◽  
Function Spaces ◽  
Dimensional Case ◽  
One Dimensional ◽  
Littlewood Theorem ◽  
The One

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

scholarly journals Approximation of a Function in Hölder Class Using Double Karamata (Kλ,μ) Method

2020 ◽  
Vol 13 (3) ◽  
pp. 567-578
Author(s):  
H. K. Nigam ◽  
Md. Hadish
Keyword(s):  
Fourier Series ◽  
Best Approximation ◽  
Double Fourier Series ◽  
Hölder Class ◽  
The Best Approximation ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Approximation Of A Function

In this paper, we establish a new theorem on the best approximation of a function of two variables belonging to H ̈older class by double Karamata (Kλ,μ) means of its double Fourier series.

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