Characterizations of Lp(R) using tight wavelet frames

In this paper, a Littlewood-Paley function characterization of the spaces L _p (R), 1 < p< ∞ is first established by means of the equivalent conditions of tight wavelet frames, wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles. With the above characterization, another characterization of L _p (R), 1 < p< ∞ is also established in terms of the weighted l ₂-norm of the wavelet frame coefficients, which can be a useful tool in harmonic analysis, approximation theory, and image processing.