  The Special Affine Fourier Transformation or the SAFT generalizes a number of
well known unitary transformations as well as signal processing and optics
related mathematical operations. Shift-invariant spaces also play an important
role in sampling theory, multiresolution analysis, and many other areas of
signal and image processing. Shannon's sampling theorem, which is at the heart
of modern digital communications, is a special case of sampling in
shift-invariant spaces. Furthermore, it is well known that the Poisson
summation formula is equivalent to the sampling theorem and that the Zak
transform is closely connected to the sampling theorem and the Poisson
summation formula. These results have been known to hold in the Fourier
transform domain for decades and were recently shown to hold in the Fractional
Fourier transform domain by A. Bhandari and A. Zayed.
  The main goal of this article is to show that these results also hold true in
the SAFT domain. We provide a short, self-contained proof of Shannon's theorem
for functions bandlimited in the SAFT domain and then show that sampling in the
SAFT domain is equivalent to orthogonal projection of functions onto a subspace
of bandlimited basis associated with the SAFT domain. This interpretation of
sampling leads to least-squares optimal sampling theorem. Furthermore, we show
that this approximation procedure is linked with convolution and semi-discrete
convolution operators that are associated with the SAFT domain. We conclude the
article with an application of fractional delay filtering of SAFT bandlimited
functions.
