A general framework for sampling and reconstruction in function spaces associated with fractional Fourier transform

The fractional Fourier transform (FRFT) has proven to be a powerful tool in optics and signal processing. Sampling theory of this transform for band-limited signals has blossomed in recent years. However, real-world signals are often not band-limited. In this paper, we first develop the theory of frames for function spaces associated with the FRFT, then we propose a general framework for FRFT-based sampling and reconstruction in function spaces without band-limiting constraints. Based upon the proposed framework, a simple necessary and sufficient condition for FRFT-based uniform sampling in function spaces is found, which facilitates a straightforward derivation of the uniform sampling theorem for the FRFT. The theoretical derivations are validated by the means of numerical results.