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The ancient Greeks went to war when someone discovered numbers which couldn’t be written as a fraction. Wait until they find out there are functions in L^2 which can’t be expressed as a Fourier integral!

Is there an algebraic structure which fully generalizes the algebra of tempered distributions with respect to both convolutions against L^p, differentiation, multiplication by functions of polynomial growth, and other operations I’m not thinking of?

The word “shadow” should be used more in mathematical terminology.

Abel summation gives meaning to divergent series and improper integrals. If we take the Fourier transform of the situation, you'll see this is equivalent to finding the boundary values of a harmonic function that doesn't necessarily have a continuous extension!

Weak convergence of boundary conditions leads to pointwise convergence of the corresponding solutions almost everywhere. If you’re lucky (or made stronger assumptions), you might get more!

Hölder’s Inequality is a consequence of the Central Limit Theorem. Pass it on.

A parameteized knot is just a periodic vector-valued function. Therefore, the theory of Fourier series applies.