Let {ψj,k}(jk)∈ℤ2 and {ψ˜j,k}(jk)∈ℤ2 be dual wavelet frames in L₂(ℝ), let η be an even, bounded, decreasing on [0, ∞) function such that∫0∞η(x)log(1+x)dx<∞, and let |ψ(x)|, |ψ˜(x)|≤η(x). Then the series ∑j,k∈ℤ(fψ˜j,k)ψj,k converges unconditionally in L_p(ℝ), 1 < p < ∞.