A wavelet-like unconditional basis. (English) Zbl 0858.42020
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.
MSC:
42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
46B04 | Isometric theory of Banach spaces |
46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |