The Wavelet Representations

Abstract

In this chapter we will discuss the presence of the Gibbs phenomenon in the continuous wavelet integral representation as well as the discrete wavelet series expansion of functions. The latter is seen more often since it represents, as we shall see shortly, an algorithm that is very efficient to compute. We will show that, to date, aside from the well known and oldest Haar wavelet, as a wavelet with jump discontinuities, almost all the well known continuous wavelets exhibit a Gibbs-type phenomenon in their approximation of functions with jump discontinuities. In general, the resulting overshoots and undershoots are smaller in magnitude than those of the Fourier integral and series representations covered in Chapters 1 and 2, and that of the general orthogonal series expansion in Chapter 3. Also, it is shown that some wavelets can be found that exhibit no such overshoots or undershoots, i.e., no Gibbs phenomenon. Indeed, for the well known wavelets the number of extremas, in general, are very few, and sometimes just only one overshoot and one undershoot on each side of the jump discontinuity. We shall discuss and illustrate this fact with complete details for the case of using the Mexican hat wavelet, and with some details for a group of Hardy functions-type wavelets (or the Poisson wavelets,) in the representation of the unit step function u(t) of Fig. 1.8b with its jump discontinuity at the origin.