Summary: We show that the orthogonal wavelet basis suitably normalized under the Wiener condition is an unconditional basis in \(L^p(\mathbb{R})\). This basis is a bounded Besselian basis in the space for \(1<p\leq 2\), and a bounded Hilbertian basis for \(2\leq p<\infty\). Under the Wiener condition, we show that the pre-wavelet basis with stable shift is a frame and we also construct the dual frame. We show that this frame provides an unconditional basis in \(L^p(\mathbb{R})\) \((1<p<\infty)\) using the Calderón-Zygmund operator.