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.

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.

Erratum: Pointwise Convergence of Fourier Series

1997 ◽  
Vol 146 (1) ◽  
pp. 239 ◽  
Author(s):  
Charles L. Fefferman
Keyword(s):  
Fourier Series ◽  
Pointwise Convergence

scholarly journals On the series of Legendre

1918 ◽  
Vol 94 (660) ◽  
pp. 292-295
Keyword(s):  
Fourier Series ◽  
General Type ◽  
General Term ◽  
Present Communication ◽  
Legendre Series ◽  
Normal Functions ◽  
Sturm Liouville ◽  
To Come ◽  
In Virtue Of

In recent communications to the Society, I have confined myself largely to the Theory of Fourier Series, partly because much seemed to me still to require doing in this subject, partly because I believed its thorough investigation to be the natural preparation for the study of other series of normal functions. It has, indeed, been known for some time that the behaviour of, for instance, series of Sturm-Liouville functions exactly corresponds to that of Fourier series. The introduction that I have recently made into Analysis of what I have called restricted Fourier series enables us to notably extend the range of such analogies. I propose in the present communication to illustrate this remark with reference to series of Legendre coefficients. Whereas Fourier series may be said to be “naturally unrestricted,” in virtue of the fact that the convergence of the integrated series to an integral necessarily involves the tendency towards zero of its own general term, so that the consideration of the more general type of series does not at once suggest itself, Legendre series may be said to come into being “restricted,” even when the coefficients are expressible in what may be called the Fourier form by means of integrals involving Legendre’s coefficients. In other words, such series correspond precisely to restricted Fourier series, instead of to ordinary Fourier series like the analogous series of Sturm-Liouville functions.

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