Unconditional convergence for wavelet frame expansions (Q1615070)

The author shows that if \(\{\psi_{j,k}\}_{(j,k)\in \mathbb Z^2}\) and \(\{\tilde\psi_{j,k}\}_{(j,k)\in \mathbb Z^2}\) (where \(f_{j,k}=2^{j/2}f(2^j\cdot-k)\)) are dual frames in \(L_2(\mathbb R)\), (i.e. there are numbers \(A, B>0\) such that \(A\|f\|^2 \leq \sum_{(j,k)\in \mathbb Z^2} (|(f,\psi_{j,k})|^2+|(f,\tilde\psi_{j,k})|^2) \leq B\|f\|^2\) and \(f= \sum_{(j,k)\in \mathbb Z^2} (f,\tilde \psi_{j,k}) \psi_{j,k}\) for every \(f\in L_2(\mathbb R)\)), then the series \(\sum_{(j,k)\in \mathbb Z^2} (f,\tilde \psi_{j,k}) \psi_{j,k}\) converges unconditionally in \(L_p(\mathbb R)\), for every \(f\in L_p(\mathbb R)\), \(1<p<\infty\), provided there is an even, bounded, and decreasing function \(\eta\) on \([0,\infty)\) such that \(\int_0^\infty \eta(x) \log(1+x)\, dx <\infty\) and \(|\psi(x)|\), \( |\tilde \psi(x)|\leq \eta(x)\).