WAVELETS AND PREWAVELETS IN LOW DIMENSIONS

In Riemenschneider and Shen (in "Approximation Theory and Functional Analysis" (C. K. Chui, Ed.), pp. 133-149, Academic Press, New York, 1991) an explicit orthonormal basis of wavelets for L2(Rs), s=1,2,3, was constructed from a multiresolution approximation given by box splines. In other words, L2(Rs) has the orthogonal decomposition ⊕ Wν. (*) ν ε{lunate} Z Orthonormal bases for the spaces Wν, are given by {2 νs 2K, j∈Zs, μ ε{lunate} Z2s{minus 45 degree rule} 0, where Zs2 and the "wavelets" Kμ are 2s - 1 cardinal splines with exponential decay. In this paper, we consider multiresolutions generated by suitable compactly supported and symmetric functions θ{symbol} and explicitly construct 2s - 1 compactly supported functions θ{symbol}μ, μ ε{lunate} Z2s{minus 45 degree rule} 0, such that the translates θ{symbol}μ(· - j), j∈Zs, are an unconditional basis for W0. Thus, the functions θ{symbol}μ(2ν· - j), ν ε{lunate} Z, j∈Zs, μ ε{lunate} Z2s {minus 45 degree rule} 0 comprise a basis for the orthogonal decomposition (*) (the functions are orthogonal for different ν because the decomposition is orthogonal, but neither the translates nor the functions will be orthogonal for given ν). The functions are given as θ{symbol}μ(·/2) 2s = μ *′ βμ with the sequences βμ formed from a single sequence by translation and change in sign pattern. We also discuss various ways to regain some of the orthogonality lost by requiring compact support. © 1992.