Unconditional Convergence for Wavelet Frame Expansions

Abstract

Let \( {\left\{{\psi}_{j,k}\right\}}_{\left(j,k\right)\in {\mathrm{\mathbb{Z}}}^2} \) and \( {\left\{{\tilde{\psi}}_{j,k}\right\}}_{\left(j,k\right)\in {\mathrm{\mathbb{Z}}}^2} \) be dual wavelet frames in L₂(ℝ), let η be an even, bounded, decreasing on [0, ∞) function such that

\( \underset{0}{\overset{\infty }{\int }}\eta (x)\log \left(1+x\right) dx<\infty, \)

and let |ψ(x)|, \( \left|\tilde{\psi}(x)\right|\le \eta (x) \). Then the series \( \sum \limits_{j,k\in \mathrm{\mathbb{Z}}}\left(f,{\tilde{\psi}}_{j,k}\right){\psi}_{j,k} \) converges unconditionally in L_p(ℝ), 1 < p < ∞.