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2013 | 11 | 7 | 1317-1333
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A gradient-projective basis of compactly supported wavelets in dimension n > 1

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A given set W = {W X } of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm \(\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} \) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W x } of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure \(\chi = (\tilde \chi ,\nu )\) , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by \(\tilde \chi \) are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by \(\tilde \chi \) are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)
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  • Mathematics Department, Texas A&M University, College Station, Texas, 77843-3368, USA
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