On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series

1978 ◽  
Vol 4 (4) ◽  
pp. 269-287 ◽  
Author(s):  
B. I. Golubov
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Spherical Means ◽  
Fourier Integrals

Investigation of discontinuous periodic signals using the measuring labview complex

2020 ◽  
pp. 72-79
Author(s):  
E.B. Solovyeva ◽  
◽  
Yu.M. Inshakov ◽  
Keyword(s):  
Fourier Series ◽  
Laboratory Experiment ◽  
Gibbs Phenomenon ◽  
Electrical Signals ◽  
Measuring Complex ◽  
Universal Measuring ◽  
Depth Study ◽  
Periodic Signals ◽  
Truncated Fourier Series

General approaches to the analysis of the Gibbs phenomenon for discontinuous periodic signals approximated by the truncated Fourier series are considered. Methods for smoothing the truncated Fourier series and improving its convergence are discussed. The software means for modeling is a universal measuring complex LabVIEW, which possesses a convenient environment for analyzing electrical signals, on the basis of this complex a laboratory experiment is carried out. The advantages of the measuring LabVIEW complex and its capabilities for in-depth study of discontinuous periodic signals are noted.

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

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.

A case of distinction between Fourier integrals and Fourier series

1927 ◽  
Vol 23 (7) ◽  
pp. 755-767
Author(s):  
Margaret Eleanor Grimshaw
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Generating Function ◽  
Fourier Integral ◽  
Finite Range ◽  
Fourier Integrals

A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.

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