Abstract: A special abstract function L²⁽R,H) is discussed, the continuous wavelet transform on L²(R) is generalized to that on L²(R,H), and the reconstruct formula produced by the integral kernel of the abstract transform function is discussed. The conditions which make the reconstrcution formula true not only in the weak topology but also in the strong topology are discussed. Generally speaking, there does not exist the reconstruction formula. The condition of abstract functions which makes the reconstruction formula the simplest is obtained. A method of finding the abstract functions which make the reconstruction formula the simplest is given.

Key words: Fourier transform, continuous wavelet transform, weak topology, strong topology