On the pointwise convergence of the continuous wavelet transforms of Lp-functions

Abstract

Under the minimal conditions on wavelets convergence almost-everywhere of wavelet transforms of Lp-functions (p>1) is well known. But this result is not completely satisfying for the reason, that we have no information about the exceptional set (of measure zero), where there is no convergence. In this paper under the slightly stronger conditions on wavelets we prove convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. On the other hand, practically all the wavelets, like Haar and "French hat" wavelets, used frequently in applications, satisfy our conditions. We also prove that the same conditions on wavelets guarantee the Riemann localization principle in L1 for the wavelet transforms.