Some simple Haar-type wavelets in higher dimensions

Abstract

An orthonormal wavelet system in ℝ^d, d ∈ ℕ, is a countable collection of functions {ψ ^ℓ_j,k }, j ∈ ℤ, k ∈ ℤ^d, ℓ = 1,..., L, of the form$\psi _{j,k}^\ell (x) = |\det a|^{ - j/2} \psi ^\ell (a^{ - j} x - k) \equiv (D_a jT_k \psi ^\ell )(x)$ that is an orthonormal basis for L² (ℝ^d), where a ∈ GL_d (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ¹(x) = ψ(x) = κ[0,1/2)^(x) − κ_[l/2,1) ^(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x₁, x₂, ..., x_d) = φ₁ (x₁)φ₂(x₂) ... φ_d(x_d) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = ( ^{1 1}_1-1 ) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form a^jb, j ∈ ℤ, where a ∈ GL_d(ℝ) has some “expanding” property and b belongs to a group of matrices in GL_d(ℝ) having ¦det b¦ = 1.