Stability of wavelet frames with matrix dilations

Abstract

Under certain assumptions we show that a wavelet frame \[ { τ ( A j , b j , k ) ψ } j , k ∈ Z := { | det A j | − 1 / 2 ψ ( A j − 1 ( x − b j , k ) ) } j , k ∈ Z \{\tau (A_j,b_{j,k})\psi \}_{j,k\in \mathbb {Z}}:= \{|\det A_j |^{-1/2} \psi (A_j^{-1}(x-b_{j,k}))\}_{j,k\in \mathbb {Z}} \] in L 2 ( R d ) L^2(\mathbb {R}^d) remains a frame when the dilation matrices A j A_j and the translation parameters b j , k b_{j,k} are perturbed. As a special case of our result, we obtain that if { τ ( A j , A j B n ) ψ } j ∈ Z , n ∈ Z d \{\tau (A^j,A^jBn)\psi \}_{j\in \mathbb {Z},n\in \mathbb {Z}^d} is a frame for an expansive matrix A A and an invertible matrix B B , then { τ ( A j ′ , A j B λ n ) ψ } j ∈ Z , n ∈ Z d \{\tau (A_j^\prime ,A^jB\lambda _n)\psi \}_{j\in \mathbb {Z}, n\in \mathbb {Z}^d} is a frame if ‖ A − j A j ′ − I ‖ 2 ≤ ε \|A^{-j}A’_j - I\|_2\le \varepsilon and ‖ λ n − n ‖ ∞ ≤ η \|\lambda _n - n\|_{\infty } \le \eta for sufficiently small ε , η > 0 \varepsilon , \eta >0 .