A perturbed Whittaker–Kotelʼnikov–Shannon sampling theorem

The sampling theorem of Whittaker (1915) [31], Kotelʼnikov (1933) [25] and Shannon (1949) [28] gives cardinal series representations for finite L 2 -Fourier transforms at equidistant sampling points. Here we investigate the situation when the Fourier transform is replaced by a perturbed one. Thus the kernel of the transform will be of the form exp ( − i x t ) + ε ( x , t ) , instead of exp ( − i x t ) in the unperturbed case. The perturbed kernel arises from first order eigenvalue problems with rank one perturbations.