Pointwise Convergence of the Calderón Reproducing Formula

Abstract

In this paper, we study the pointwise convergence of the Calderon reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every \(f\in L_{w}^{p}(\mathbb {R}^{d})\) with an \(\mathcal{A}_{p}\) weight w, 1≤p<∞, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.