Norms of powers of absolutely convergent fourier series

1998 ◽  
Vol 76 (1) ◽  
pp. 137-161 ◽  
Author(s):  
Bogdan M. Baishanski ◽  
Michael R. Snell
Keyword(s):  
Fourier Series ◽  
Convergent Fourier Series

On Banach spaces of absolutely and strongly convergent Fourier series

1990 ◽  
Vol 55 (1-2) ◽  
pp. 149-160 ◽  
Author(s):  
I. Szalay ◽  
N. Tanović-Miller
Keyword(s):  
Fourier Series ◽  
Banach Spaces ◽  
Convergent Fourier Series

Almost Everywhere Convergent Fourier Series

2011 ◽  
Vol 18 (2) ◽  
pp. 266-286 ◽  
Author(s):  
M. J. Carro ◽  
M. Mastyło ◽  
L. Rodríguez-Piazza
Keyword(s):  
Fourier Series ◽  
Convergent Fourier Series ◽  
Almost Everywhere

scholarly journals Corrected Fourier series and its application to function approximation

2005 ◽  
Vol 2005 (1) ◽  
pp. 33-42 ◽  
Author(s):  
Qing-Hua Zhang ◽  
Shuiming Chen ◽  
Yuanyuan Qu
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Function Approximation ◽  
Gibbs Phenomenon ◽  
Correction Function ◽  
Algebraic Polynomials ◽  
Convergent Fourier Series ◽  
Step Functions ◽  
Uniformly Convergent ◽  
Uniformly Convergent Fourier Series

Any quasismooth functionf(x)in a finite interval[0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e.,f(0)≠f(x0)) and the finite discontinuities in between. The uniformly convergent Fourier series and the correction function are collectively referred to as the corrected Fourier series. We prove that in order for themth derivative of the Fourier series to be uniformly convergent, the order of the polynomial need not exceed(m+1). In other words, including the no-more-than-(m+1)polynomial has eliminated the Gibbs phenomenon of the Fourier series until itsmth derivative. The corrected Fourier series is then applied to function approximation; the procedures to determine the coefficients of the corrected Fourier series are illustrated in detail using examples.

scholarly journals Absolutely convergent Fourier series and generalized Zygmund classes of functions

2009 ◽  
Vol 104 (1) ◽  
pp. 124
Author(s):  
Ferenc Móricz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Fourier Coefficients ◽  
Sufficient Conditions ◽  
Modulus Of Smoothness ◽  
Slowly Varying Function ◽  
Continuous Periodic Function ◽  
Convergent Fourier Series ◽  
Period 2 ◽  
Order Of Magnitude

We investigate the order of magnitude of the modulus of smoothness of a function $f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belongs to one of the generalized Zygmund classes $(\mathrm{Zyg}(\alpha, L)$ and $(\mathrm{Zyg} (\alpha, 1/L)$, where $0\le \alpha\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) \to \infty$ as $x\to \infty$. A continuous periodic function $f$ with period $2\pi$ is said to belong to the class $(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all $x\in \mathsf T$ and $h>0$}, 26740 where the constant $C$ does not depend on $x$ and $h$; and the class $(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.

scholarly journals Some Fourier division problems

1971 ◽  
Vol 12 (2) ◽  
pp. 167-186
Author(s):  
R. E. Edwards
Keyword(s):  
Fourier Series ◽  
Convergent Fourier Series ◽  
Circle Group ◽  
Division Problems ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued

Let T denote the circle group, C the set of continuous complex-valued functions on T, and A the set of f ∈ C having absolutely convergent Fourier series:I standing for the set of integers.

scholarly journals On Grasia's criterion for uniform convergence of Fourier series

1991 ◽  
Vol 51 (2) ◽  
pp. 305-323
Author(s):  
Charles Oehring
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Besov Space ◽  
Simple Proof ◽  
Sharp Estimate ◽  
Embedding Theorem ◽  
Convergent Fourier Series ◽  
Uniformly Convergent ◽  
Uniformly Convergent Fourier Series

AbstractGarsia's discovery that functions in the periodic Besov space λ(p-1,p, 1), with 1 <p< ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1,p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α,p, q)

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