scholarly journals On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series

2021 ◽  
Vol 25 (1) ◽  
pp. 5-19
Author(s):  
Xhevat Krasniqi
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Bounded Variation ◽  
Degree Of Approximation ◽  
Continuous Functions ◽  
Auxiliary Function ◽  
Partial Sums ◽  
The Mean ◽  
Approximation Of Continuous Functions ◽  
Positive Integers

Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}∞n=1.

scholarly journals The approximation of continuous functions by Riesz typical means of their Fourier series

1967 ◽  
Vol 7 (4) ◽  
pp. 539-544 ◽  
Author(s):  
B. Kwee
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Continuous Functions ◽  
Positive Order ◽  
Approximation Of Continuous Functions

Let (x) be a continuous function with period 2π. It is well known that the Fourier series of (x) is summable Riesz of any positive order to (x). The aim of this paper is the proof of the following theorem.

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.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals A Note on the Approximation of Continuous Functions by Riesz Typical Means of Their Fourier Series

1969 ◽  
Vol 9 (1-2) ◽  
pp. 180-181 ◽  
Author(s):  
B. Kwee
Keyword(s):  
Fourier Series ◽  
Positive Integer ◽  
Continuous Functions ◽  
Image Position ◽  
Approximation Of Continuous Functions

In [1], the following theorem is proved:THEOREM. If f ∈C2π, α is a positive integer, and then Where , .

Convergence of the Hausdorff Means of Double Fourier Series*

1968 ◽  
Vol 11 (4) ◽  
pp. 585-591 ◽  
Author(s):  
Fred Ustina
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Double Fourier Series ◽  
Partial Sums ◽  
Closed Region ◽  
Hausdorff Means

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

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