Gibbs' phenomenon for sampling series and what to do about it

Abstract

ABSTRACT. Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.

The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

“…This is in addition to the recent active research for the Gibbs phenomenon in most of the continuous wavelets integral representations [35,36,37]' and the discrete wavelets series expansions [32,[38][39][40]30]. This research is combined with attempts at reducing the Gibbs phenomenon, centering around the Fejer averaging or the Cesaro summability [38][39][40], and our most recent Lanczos-like local filtering [41,22].…”

“…In the short period since 1992, more references have been found for the Gibbs Phenomenon in Fourier-Bessel series expansion [23][24][25][26][27], in Fourier series in higher dimensions [25,27]' in spline expansions [28][29][30][31][32], in the interpolation of the discrete Fourier transform (DFT) [33], and in how it depends on the way we measure it as an error (in Lp-approximation, for example) [34]. This is in addition to the recent active research for the Gibbs phenomenon in most of the continuous wavelets integral representations [35,36,37]' and the discrete wavelets series expansions [32,[38][39][40]30]. This research is combined with attempts at reducing the Gibbs phenomenon, centering around the Fejer averaging or the Cesaro summability [38][39][40], and our most recent Lanczos-like local filtering [41,22].…”

Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis

“…The part of the curve for which Tmf(x) > f(x) ( Recently, Walter and Shim [13] found a necessary and sufficient condition for the existence of the Gibbs ripple in the sampling wavelet expansions of such functions. Moreover, they gave examples of wavelet sampling series which do not exhibit Gibbs Phenomenon.…”

“…For the pointwise convergence, Tmf(X) ~ f(x) as m --~ ec, we refer to [1], [2], [4], and [13]. For the behavior of Tmf near a jump discontinuity of f, we refer to [1], [2], and [5].…”

Gibbs’ phenomenon in higher dimensions

“…It has been shown to exists for many natural approximations, e.g., those involving Fourier series and other classical orthogonal expansions [14], orthogonal and sampling series of wavelets [8,16,20,22], various approximations to integral transforms [7]. A few attempts to extend the concept to higher dimensions have been made, but have considered only Fourier approximations and special partial sums [6,[11][12][13]19,21,23].…”