Stability of wavelet frames with matrix dilations

Under certain assumptions we show that a wavelet frame

in $L^2(\mathbb{R} ^d)$ remains a frame when the dilation matrices $A_j$ and the translation parameters $b_{j,k}$ are perturbed. As a special case of our result, we obtain that if $\{\tau(A^j,A^jBn)\psi\}_{j\in \mathbb{Z} ,n\in \mathbb{Z} ^d}$ is a frame for an expansive matrix $A$ and an invertible matrix $B$, then $\{\tau(A_j^\prime,A^jB\lambda_n)\psi\}_{j\in \mathbb{Z} , n\in\mathbb{Z} ^d}$ is a frame if $\Vert A^{-j}A'_j - I\Vert _2\le \varepsilon$and $\Vert\lambda_n - n\Vert _{\infty} \le \eta$ for sufficiently small $\varepsilon, \eta>0$.