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A note on constructing affine systems for L2

By H.-Q. Bui, N. Kaiblinger and R.S. Laugesen

Abstract

AbstractAssume ψ∈L2(Rd) has Fourier transform continuous at the origin, with ψˆ(0)=1, and that ∑l∈Zd|ψˆ(ξ−l)|2 is bounded as a function of ξ∈Rd. Then every function f∈L2(Rd) can be represented by an affine series f=∑j>0∑k∈Zdcj,kψj,k for some coefficients satisfying‖c‖ℓ1(ℓ2)=∑j>0(∑k∈Zd|cj,k|2)1/2<∞. Here ψj,k(x)=|detaj|1/2ψ(ajx−k) and the dilation matrices aj expand, for example aj=2jI. The result improves an observation by Daubechies that the linear combinations of the ψj,k are dense in L2(Rd)

Publisher: Elsevier Inc.
Year: 2008
DOI identifier: 10.1016/j.acha.2008.04.001
OAI identifier:

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