On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

Abstract

Gibbs effect represents the non-uniform convergence of the nth Fourier partial sums in approximating functions in the neighborhood of their non-removable discontinuities (jump discontinuities). The overshoots and undershoots cannot be removed by adding more terms in the series. This effect has been studied in the literature for wavelet and framelet expansions. Dual tight framelets have been proven useful in signal processing and many other applications where translation invariance, or the resulting redundancy, is very important. In this paper, we will study this effect using the dual tight framelets system. This system is generated by the mixed oblique extension principle. We investigate the existence of the Gibbs effect in the truncated expansion of a given function by using some dual tight framelets representation. We also give some examples to illustrate the results.