Upper bounds for Fourier transforms of exponential functions

Meaningful upper bounds for the Fourier transform of polynomial exponential functions are often hard to come by. Regarding Fourier transforms of rational exponential functions, which are of importance, for example in Campbell's sampling theorem, the purpose of finding significant upper bounds is an even more demanding exercise. In this article, we propose a new approach in order to obtain significant upper bounds for Fourier transforms of general exponential functions. The technique is shown to allow further generalization in order to deal with Fourier-like integrals and rational exponential integrals.