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.

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

scholarly journals The double Fourier series of a discontinuous function

1924 ◽  
Vol 106 (737) ◽  
pp. 299-314 ◽  
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Double Fourier Series ◽  
General Term ◽  
Discontinuous Function ◽  
Lebesgue Integral ◽  
Function Of Two Variables

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function f ( x, y ) possesses a double Lebesgue integral over the square (– π < π ; – π < y < π ). Then the general term of the double Fourier series of this function is given by cos = є mn { a mn cos mx cos ny + b mn sin mx sin ny + c mn cos mx sin ny + d mn sin mx cos ny } There є 00 = ¼, є m0 = ½ ( m > 0), є 0n = ½ ( n > 0), є ms = 1 ( m > 0, n >0). the coefficients are given by the formulæ a mn = 1/ π 2 ∫ π -π ∫ π -π f ( x, y ) cos mx cos ny dx dy , obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫ ms = Σ m μ = 0 Σ n v = 0 A μ v = 1/π 2 ∫ π -π ∫ π -π f (s, t) sin( m +½) ( s - x ) sin ( n + ½) ( t - y )/2 sin ½ ( s - x ) 2 sin ½ ( t - y ) if f ( s , t ) is defined outside the original square by double periodicity, we have sub S ms = 1/π 2 ∫ π 0 ∫ π 0 f ( x + s , y + t ) + f ( x + s , y - t ) + f ( x - s , y + t ) + f ( x - s , y - t ) sin ( m + ½) s / 2 sin ½ s sin ( n + ½) t / 2 sin ½ t ds dt .

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 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 Functions in Besov Space

2021 ◽  
Vol 52 ◽  
Author(s):  
Hare Krishna Nigam ◽  
Supriya Rani
Keyword(s):  
Fourier Series ◽  
Besov Space ◽  
Best Approximation ◽  
Approximation Of Functions ◽  
Approximation Of A Function

In the present paper, we establish a theorem on best approximation of a function g ∈ Bqλ(Lr) of its Fourier series. Our main theorem generalizes some known results of this direction of work. Thus, the results of [10], [26] and [27] become the particular case of our main Theorem 3.1.

scholarly journals Approximation of Function in generalized Hölder Class

2020 ◽  
Vol 13 (2) ◽  
pp. 351-368
Author(s):  
Supriya Rani ◽  
H. K. Nigam
Keyword(s):  
Fourier Series ◽  
Error Estimation ◽  
Hölder Class ◽  
Holder Class ◽  
Approximation Of Function

IIn the present work, we study error estimation of a function g ∈ H(η) r (r ≥ 1) class using Matrix-Hausdorff (T ∆H) means of its Fourier series. Our Theorem 1 generalizes twelve previously known results. Thus, the results of [4-5, 11–16, 18, 26, 29-30] become the particular cases of our Theorem 1. Several useful results in the form of corollaries are also deduced from our Theorem 1.

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