Wavelets and Framelets Within the Framework of Nonhomogeneous Wavelet Systems

Abstract

In this paper, we shall discuss recent developments in the basic theory of wavelets and framelets within the framework of nonhomogeneous wavelet systems in a natural and simple way. We shall see that nonhomogeneous wavelet systems naturally link many aspects of wavelet analysis together. There are two fundamental issues of the basic theory of wavelets and framelets: frequency-based nonhomogeneous dual framelets in the distribution space and stability of nonhomogeneous wavelet systems in a general function space. For example, without any a priori condition, we show that every dual framelet filter bank derived via the oblique extension principle (OEP) always has an underlying frequency-based nonhomogeneous dual framelet in the distribution space. We show that directional representations to capture edge singularities in high dimensions can be easily achieved by constructing nonstationary nonhomogeneous tight framelets in $${L}_{2}(\mathbb{R})$$ with the dilation matrix $$2{I}_{mathbbD}$$ . Moreover, such directional tight framelets are derived from tight framelet filter banks derived via OEP. We also address the algorithmic aspects of wavelets and framelets such as discrete wavelet/framelet transform and its basic properties in the discrete sequence setting. We provide the reader in this paper a more or less complete picture so far on wavelets and framelets with the framework of nonhomogeneous wavelet systems.