On the Regularization of Divergent Fourier Transforms

Derivation of a way to take the Fourier transforms over the entire real line of functions for which the usual definition diverges, specifically: $\mathcal{F}\left\{f\left(e^{-t}\right)\right\}_s$ for $f\colon\mathcal{R}^{+}\xrightarrow{\sim}\mathbb{R}$.