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 ψ̂ = χK, is an orthonormal wavelet for L2(Rn). Here we provide methods for constructing explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional counterparts.
Soardi, P., Weiland, D. (1998). Single Wavelets in n-Dimensions. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 4(3), 299-315 [10.1007/BF02476029].
Single Wavelets in n-Dimensions
SOARDI, PAOLO MAURIZIO;
1998
Abstract
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 ψ̂ = χK, is an orthonormal wavelet for L2(Rn). Here we provide methods for constructing explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional counterparts.
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