Generalized comb function: a new self-Fourier function

The comb function is defined as equidistantly spaced impulses (i.e., an impulse train); it is well known that its Fourier transform also provides a comb function, and is used for a proof of the sampling theorem. As a generalization of this function, we propose a novel comb function, called "generalized comb function" (GCF), that consists of equally spaced but proportionally expanded pulses along the trans-versal axis. It is shown that the Fourier transform of the GCF with an arbitrary pulse shape can be obtained only by replacement of variables without any Fourier integral operation, and the transformed function is also included in the GCF family, like that of the conventional comb function. The theorem representing this relationship and some examples are presented.