A note on constructing affine systems for L2

Assume 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).