Affine synthesis in the space L2(Rd)

We establish some theorems on the representation of functions f element of L2(Rd) by series of the form f=ΣjelementofNΣkelementofZdcj,kψj,k that are absolutely convergent with respect to the index j (that is, ΣjelementofN||ΣkelementofZdcj,kψj,k||2<∞), where ψj,k(x)=|det aj|1/2ψ(ajx-bk), j element of N, k element of Zd, is an affine system of functions. We prove the validity of the Bui-Laugesen conjecture on the sufficiency of the Daubechies conditions for a positive solution of the affine synthesis problem in the space L2(Rd). A constructive solution is given for this problem under a localization of the Daubechies conditions