Orthonormal sampling functions

We investigate functions ϕ(x) whose translates {ϕ(x−k)}, where k runs through the integer lattice Z, provide a system of orthonormal sampling functions. The cardinal sine, whose important role in the sampling theory of bandlimited functions is well documented, is the classic example. For the bandlimited case we provide a complete characterization of such functions ϕ and give examples with rapid decay including a construction which is symmetric. We also analyze the general case of arbitrary sampling rate, a>0, which leads to some unexpected observations.