A note on sampling expansion for a transform with parabolic cylinder kernel

Abstract

This note concerns a sampling expansion for integral transforms, of functions defined on the half line, with a parabolic cylinder (Hermite-Weber) function as its kernel. The sampling with such transforms for functions on the whole real line is not possible, as the parabolic cylinder function is not square integrable there and the sampling function does not exist in the usual finite form. This should set straight some unsuccessful attempts, concerning such expansions, which appeared in this journal.