Higher-Dimensional Wavelet Systems

Abstract

In this chapter, we construct compactly supported (orthonormal) systems of wavelets for L²(R^d), d > 1. Each system is of the form

$$\left\{ {{\psi ^s}\left( {{M^k}x - p} \right):1 \le s \le m - 1;k \in Z;p \in \Lambda } \right\},$$

(7.1)

where Λ is a rank d lattice subgroup of R^d, M is a strictly expansive linear transformation with MΛ ⊂ Λ, and m = |detM|. The fundamental wavelets, ψ^s, 1 ≤ s ≤ m −1 are defined by functional equations having the form

$${\psi ^s}\left( x \right) = \sum\limits_{p \in \Lambda } {a_p^s\varphi \left( {Mx - p} \right),\,\,1 \le s \le m - 1,} $$

(7.2)

where a ^s _p are complex-valued functions having finite support on Λ and satisfy certain algebraic conditions, and the scaling function ϕ satisfies the scaling equation

$$\varphi \left( x \right) = \sum\limits_{p \in \Lambda } {a_p^0\varphi \left( {Mx - p} \right).}$$