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

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Battle G.

Open Mathematics

2013

tom 11

nr 7

1317-1333

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.)

Ograniczanie wyników

1 Open Mathematics

1 Battle G.

1 2013

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