Generalized Spectrum of Interpolants: Part I - Theory

In this paper, we propose a new theorem relating to the generalized spectrum of an interpolating function. In particular, our theorem provides an incite to design interpolants in frequency domain. We consider a more pragmatic situation where the reconstruction filter is non-ideal and occupies some extra bandwidth. The spectrum is shown to satisfy the orthogonality condition which the strongest constraint for the admissibility of an interpolant. Extending our formulation, we show that the spectrum of generating function (Sine) used in Shannon´s sampling theorem is a special case of spectrum proposed in this paper.