scholarly journals Uniformly Convergent Fourier Series and Multiplication of Functions

2018 ◽  
Vol 303 (1) ◽  
pp. 171-177
Author(s):  
V. V. Lebedev
Keyword(s):  
Fourier Series ◽  
Convergent Fourier Series ◽  
Uniformly Convergent ◽  
Uniformly Convergent Fourier Series

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 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)

scholarly journals Uniform convergence and everywhere convergence of Fourier series. I

1973 ◽  
Vol 9 (3) ◽  
pp. 321-335 ◽  
Author(s):  
Masako Izumi ◽  
Shin-ichi Izumi
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Convergent Fourier Series ◽  
Almost Everywhere ◽  
Uniformly Convergent ◽  
Uniformly Convergent Fourier Series

Carleson and Hunt proved that the space of functions with almost everywhere convergent Fourier series contains Lp (p > l) as a subspace. We shall give two kinds of subspaces of the spaces of functions with everywhere convergent or uniformly convergent Fourier series.

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