Fractional shifting and sampling in the fractional domain. Application to digital holography

Periodically shifting replicas of a function is equivalent to sampling its standard Fourier transform, but not its general fractional order Fourier transform. Indeed, sampling in the fractional domain can be achieved after defining fractional shift operators and a fractional convolution product. Standard formulae that hold true for standard operations can be generalized to the fractional calculus. A proof of the fractional sampling theorem is derived by using the former operators and the theorem is applied to computing digital holograms that work in the Fresnel regime. A condition for optimizing the distance between a diffracting object and the diffracted pattern detected by a discrete device is also given: it allows for sampling at an appropriate rate which is in accordance With the detector cell period as well as with the sampling theorem and interpolation formula.