We focus on the L^p(R²) theory of the fractional Fourier transform (FRFT) for 1 ≤ p ≤ 2. In L¹(R²), we mainly study the properties of the FRFT via introducing the two-parameter chirp operator. In order to get the point-wise convergence for the inverse FRFT, we introduce the fractional convolution and establish the corresponding approximate identities. Then the well-defined inverse FRFT is given via approximation by suitable means, such as fractional Gauss means and Able means. Furthermore, if the signal F_α,βf is received, we give the process of recovering the original signal f with MATLAB. In L²(R²), the general Plancherel theorem, direct sum decomposition, and the general Heisenberg inequality for the FRFT are obtained.