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

Mutaz Mohammad

doi:10.3390/math7100952

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.

Highlights

The Gibbs effect was first recognized over a century ago by Henry Wilbraham in 1848.in 1898 Albert Michelson and Samuel Stratton observed it via a mechanical machine that they used to calculate the Fourier partial sums of a square wave function
We will use dual tight framelets constructed by the mixed oblique extension principle (MOEP) which enables us to construct dual tight framelets for L2 (R) of the form {ψj,k, ψj,k, ` = 1, · · ·, r } j,k
We present some numerical illustration by using the dual tight framelets which will generalize the result in Ref. [14]

Summary

Introduction

The Gibbs effect was first recognized over a century ago by Henry Wilbraham in 1848 (see Ref. [1]). By considering Fourier series, it is impossible to recover accurate point values of a periodic function with many finitely jump discontinuities from its Fourier coefficients Wavelets and their generalizations (framelets) have great success in coefficients recovering and have many applications in signal processing and numerical approximations We will use dual tight framelets constructed by the mixed oblique extension principle (MOEP) It is known from the approximation theory, see e.g., Refs.

Gibbs Effect in Quasi-Affine Dual Tight Framelet Expansions

Results and Discussion

Conclusions