Single wavelets in n-dimensions

Under very minimal regularity assumptions, it can be shown that 2ⁿ−1 functions are needed to generate an orthonormal wavelet basis for L²(ℝⁿ). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝⁿ such that the single function ψ, defined by , is an orthonormal wavelet for L²(ℝⁿ). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.