Shannon-Type Wavelets and the Convergence of Their Associated Wavelet Series

Abstract

The sinc function, φ(x) = sin πx/πx, plays a crucial role in the Whittaker – Shannon – Kotel'nikov sampling theorem. This role is based on the fact that its integer translations {φ(x−n)}, form an orthonormal basis for the space of functions bandlimited to [−π,π]. In recent years, this fact has been re-explained in the context of wavelets and multiresolution analysis, namely, the sine function is a scaling function of a multiresolution analysis. Thus, it gives rise to a mother wavelet, known as the Shannon wavelet, that generates an orthonormal basis for L²ℝ. Both the sine function and Shannon wavelet have slow decay at infinity and are not integrable over the real line.

In this chapter, we introduce a class of wavelets, which we call Shannon – type wavelets, that contains the Shannon wavelet as a special case. These wavelets have many similar properties to those of the Shannon wavelet, including the poor time – frequency localization. We then investigate the point – wise convergence of wavelet series expansions of L²(ℝ) functions when the mother wavelet is of Shannon – type. It is shown that the mere continuity of a function at a point is not sufficient for its wavelet series to converge to it at that point.