Banach frames in the affine synthesis problem

We consider the problem of representing functions $f\in L^p(\mathbb R^d)$ by a series in elements of the affine system
$$ \psi_{j,k}(x)=\lvert\det a_j\rvert^{1/2}\psi(a_jx-bk), \qquad j\in\mathbb N, \quad k\in\mathbb Z^d. $$
The corresponding representation theorems are established on the basis of the frame inequalities
$$ A\|g\|_q\le\|\{(g,\psi_{j,k})\}\|_Y\le B\|g\|_q $$
for the Fourier coefficients $\displaystyle(g,\psi_{j,k})=\int_{\mathbb R^d}g(x)\psi_{j,k}(x)\,dx$ of functions $g\in L^q(\mathbb R^d)$, $1/p+1/q=1$, where ${\|\cdot\|}_Y$ is the norm in some Banach space of number families $\{y_{j,k}\}$ and $0<A\le B<\infty$ are constants.
In particular, it is proved that if the integral of a function $\psi\in L^1\cap L^p(\mathbb R^d)$, $1<p<\infty$, is nonzero, so $\displaystyle\int_{\mathbb R^d}\psi(x)\,dx\ne0$ and the system of translates $\{\psi(x-bk):k\in\mathbb Z^d\}$ is $p$-Besselian in the space $L^p(\mathbb R^d)$, then for any function $f\in L^p(\mathbb R^d)$ we have the representation
$$ f=\sum_{j\in\mathbb N}\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}, $$
where the coefficients satisfy the condition
$$ \sum_{j\in\mathbb N}\lvert\det a_j\rvert^{1/2-1/p} \biggl(\sum_{k\in\mathbb Z^d}|c_{j,k}|^p\biggr)^{1/p}<\infty. $$

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