| We introduce a new class of integral operators acting on the Hilbert space L2( R ) with an exponentially quadratic kernel and orthonormal basis of Hermite functions. We establish a one-to-one correspondence between the closed unit disk in the complex plane and this class of linear operators. We show that the usual Fourier transform pair and fractional Fourier transforms are subgroups of this class, thereby obtaining the name generalized Fourier transforms. We establish various properties of these transforms. We propose several algorithms for the numerical approximation of such transforms of suitably regular functions. Finally, we use our numerical method with Fresnel diffraction to solve a masking problem. |
