scholarly journals On the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals

2021 ◽  
Vol 103 (3) ◽  
pp. 131-139
Author(s):  
Gulsim A. Yessenbayeva ◽  
◽  
Gulmira A. Yessenbayeva ◽  
A.T. Kasimov ◽  
N.K. Syzdykova ◽  
...  
Keyword(s):  
Fourier Series ◽  
Approximation Theory ◽  
Trigonometric Polynomials ◽  
Partial Sums ◽  
Practical Applications ◽  
Noise Interference ◽  
The Best Approximation ◽  
Function Classes ◽  
Main Components ◽  
Approximation Of A Function

The article is devoted to the study of some data from the theory of functions approximation by trigonometric polynomials with a spectrum from special sets called harmonic intervals. Due to the limited perception range of devices, the perception range of the senses of the person himself, when studying a mathematical model it is often enough to find an approximation of the object so that the error (noise, interference, distortion) is outside the interval of perception. Harmonic intervals model problems of this kind to some extent. In the article the main components of the approximation theory of functions by trigonometric polynomials with a spectrum from harmonic intervals are presented, the theorem on estimating the best approximation of a function by trigonometric polynomials through the best approximations of a function by trigonometric polynomials with a spectrum from harmonic intervals is proved. Theorems on the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals are considered; such a theorem for the Lorentz space is generalized and proved. The article is mainly aimed at scientific researchers dealing with practical applications of the approximation theory of functions by trigonometric polynomials with a spectrum from special sets.

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.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

scholarly journals Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

2021 ◽  
Vol 19 (1) ◽  
pp. 1047-1055
Author(s):  
Zhihua Zhang
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Fourier Coefficients ◽  
Random Processes ◽  
Qualitative Theory ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Fourier Approximation ◽  
Random Differential Equations

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

scholarly journals Approximation of a Function Belonging to Lip ((α, r) Class by Np,q. C1 Summability Method of its Fourier Series

2014 ◽  
Vol 14 (2) ◽  
pp. 117-122 ◽  
Author(s):  
JP Kushwaha ◽  
BP Dhakal
Keyword(s):  
Fourier Series ◽  
Science And Technology ◽  
Degree Of Approximation ◽  
Summability Method ◽  
Approximation Of A Function

In this paper, an estimate for the degree of approximation of a function belonging to Lip(α, r) class by product summability method Np.q.C1 of its Fourier series has been established. DOI: http://dx.doi.org/10.3126/njst.v14i2.10424 Nepal Journal of Science and Technology Vol. 14, No. 2 (2013) 117-122

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