I tried answering a question that ended up with an expression $$\mathcal F\left\{e^{\left(\frac{2\pi j} {t}\right)}\right\}$$
Now this function we know from famous identity is $$e^{ai} = \cos(a)+i\sin(a)$$ gives $$e^{\left(\frac{2\pi j} {t}\right)} = \cos\left(\frac{2\pi} t\right)+i\sin\left(\frac{2\pi} t\right)$$
having very wobbly behaviour around $t=0$, although still being continuous and differentiable a.e.
Now to question. Would it make sense to create integral transform based on basis functions like this? Which functions could it describe well?
