Affine synthesis in the space $L^2(\mathbb R^d)$

We establish some theorems on the representation of functions $f\in L^2(\mathbb R^d)$ by series of the form $f=\sum_{j\in\mathbb N}\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}$ that are absolutely convergent with respect to the index $j$ (that is, $\sum_{j\in\mathbb N}\bigl\|\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}\bigr\|_2<\infty$), where $\psi_{j,k}(x)=|{\det a_j}|^{1/2}\psi(a_jx-bk)$, $j\in\mathbb N$, $k\in\mathbb Z^d$, 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 $L^2(\mathbb R^d)$. A constructive solution is given for this problem under a localization of the Daubechies conditions.