CONVERGENCE OF THE CONTINUOUS WAVELET TRANSFORMS ON THE ENTIRE LEBESGUE SET OF Lp-FUNCTIONS

Abstract

The almost everywhere convergence of wavelets transforms of L_p-functions under minimal conditions on wavelets is well known. But this result does not provide any information about the exceptional set (of Lebesgue measure zero), where convergence does not hold. In this paper, under slightly stronger conditions on wavelets, we prove convergence of wavelet transforms everywhere on the entire Lebesgue set of L_p-functions. On the other hand, practically all the wavelets, including 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.