# A gradient-projective basis of compactly supported wavelets in dimension n > 1

Open Mathematics (2013)

• Volume: 11, Issue: 7, page 1317-1333
• ISSN: 2391-5455

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## Abstract

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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 $\sum _{\chi }\left({\int }_{{ℝ}^{n}}\nabla f·\nabla {W}_{\chi }^{*}\right){W}_{\chi }$ converges to f with respect to the norm ${∥\nabla \left(·\right)∥}_{{L}^{2}\left({ℝ}^{n}\right)}$ . 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 $\chi =\left(\stackrel{˜}{\chi },\nu \right)$ , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by $\stackrel{˜}{\chi }$ 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 $\stackrel{˜}{\chi }$ 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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