THE APPROXIMATION OF CONTINUOUS FUNCTIONS BY PARTIAL SUMS OF ITS FOURIER SERIES

1983 ◽  
Vol 3 (4) ◽  
pp. 439-443
Author(s):  
Zhurui Guo
Keyword(s):  
Fourier Series ◽  
Continuous Functions ◽  
Partial Sums ◽  
Approximation Of Continuous Functions

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 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 , .

scholarly journals Approximation by double Walsh polynomials

1992 ◽  
Vol 15 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Ferenc Móricz
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Partial Sums ◽  
Lipschitz Classes ◽  
Sufficient Conditions For Convergence ◽  
By Products ◽  
De La Vallée Poussin

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach spaceX. In particular,Xmay beLp(I2), where1≦p<∞andI2=[0,1)×[0,1), orCW(I2), the latter being the collection of uniformlyW-continuous functions onI2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence inLp(I2)-norm and uniform convergence onI2as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.

scholarly journals Kloosterman paths and the shape of exponential sums

2016 ◽  
Vol 152 (7) ◽  
pp. 1489-1516 ◽  
Author(s):  
Emmanuel Kowalski ◽  
William F. Sawin
Keyword(s):  
Fourier Series ◽  
Exponential Sums ◽  
Continuous Functions ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Space Of Continuous Functions ◽  
Convergence In Law ◽  
Random Fourier Series ◽  
Polygonal Paths

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums$\text{Kl}_{p}(a)$, as$a$varies over$\mathbf{F}_{p}^{\times }$and as$p$tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

Sign in / Sign up

Export Citation Format

Share Document