The system of translates and the special affine Fourier transform

The translation operator $T^A$ associated with the special affine Fourier transform (SAFT) ${\mathcal{F}}_A$ is introduced from harmonic analysis point of view. The analogues of Wendel’s theorem, Wiener theorem, Weiner-Tauberian theorem and Bernstein type inequality in the context of the SAFT are established. The shift invariant space $V_A$ associated with the special affine Fourier transform is introduced and studied along with sampling problems.