Topic
Sigma approximation
About: Sigma approximation is a research topic. Over the lifetime, 44 publications have been published within this topic receiving 1143 citations. The topic is also known as: σ-approximation.
Papers
TL;DR: The Gibbs phenomenon is reviewed from a different perspective and it is shown that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case.
Abstract: The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon.
This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting.
The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation.
856 citations
TL;DR: In this paper, the Gibbs effect was examined for wavelet expansions of functions at points with jump discontinuities and certain conditions on the size of the wavelet kernel were examined to determine if a Gibbs effect occurs and what magnitude it is.
93 citations
TL;DR: An analogue of the Gibbs phenomenon is shown to hold for approximation by periodic spline functions on uniform subdivisions.
52 citations
TL;DR: In this paper, it was shown that a Gibbs phenomenon occurs in the wavelet expansion of a function with a jump discontinuity at 0 for a wide class of wavelets, and the asymptotic behavior of the Gibbs splines was analyzed.
49 citations
TL;DR: The Fourier interpolation polynomials for periodic functions with an isolated jump discontinuity exhibit for growing order a Gibbs phenomenon as discussed by the authors, however, the over-and undershots differ from the ones appearing for the partial sums of the Fourier series and depend on the coincidence of the jump with interpolation nodes.
48 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2016 | 3 |
| 2015 | 1 |
| 2014 | 2 |
| 2013 | 1 |
| 2012 | 3 |