Extensions of Multiresolution Analysis

Abstract

Multiresolution analysis (MRA) is considered as the heart of wavelet theory. The concept of MRA provides an elegant tool for the construction of wavelets. An MRA is an increasing family of closed subspaces $\left \{V _{j}: j \in \mathbb{Z}\right \}$ of $L^{2}(\mathbb{R})$ such that $\bigcap _{j\in \mathbb{Z}}V _{j} = \left \{0\right \},\,\bigcup _{j\in \mathbb{Z}}V _{j}$ is dense in $L^{2}(\mathbb{R})$ and which satisfies f ∈ V _j if and only if f(2⋅ ) ∈ V _j+1. Furthermore, there exists an element ϕ ∈ V ₀ such that the collection of integer translates of function $\upphi,\,\left \{\upphi (\cdot - k): k \in \mathbb{Z}\right \}$ represents a complete orthonormal system for V ₀. The function ϕ is called the scaling function or the father wavelet. This classic concept of MRA has been extended in various ways in recent years. These concepts are generalized to $L^{2}(\mathbb{R}^{d})$, to lattices different from $\mathbb{Z}^{d}$, allowing the subspaces of MRA to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M ≥ 2 or by an expansive matrix $A \in GL_{d}(\mathbb{R})$ as long as $A \subset A\mathbb{Z}^{d}$. From the last decade, this elegant tool for the construction of wavelet bases have been extensively studied by several authors on the various spaces, namely, abstract Hilbert spaces, locally compact Abelian groups, Cantor dyadic groups, Vilenkin groups, local fields of positive characteristic, p-adic fields, Hyrer-groups, Lie groups, zero-dimensional groups. Notice that the technique is similar to that in the real case of $\mathbb{R}$ while the mathematical treatment needs ones conscientiousness.

The wavelets arrive in succession, and each wavelet eventually dies out. The wavelets all have the same basic form and shape, but the strength or impetus of each wavelet is random and uncorrelated with the strength of the other wavelets. Despite the fore-ordained death of any individual wavelet, the time-series does not die. The reason is that a new wavelet is born each day to take the place of the one that does die on any given day, the time-series is composed of many living wavelets, all of a different age, some young, others old. Ender A. Robinson