This operational approach to a definition of Fourier/Laplace transforms has been developed by R.A. Kunze, <A HREF="http://www.jstor.org/stable/1970090">An operator theoretic approach to generalized Fourier transforms</A>.

> This is an attempt to give an intrinsic definition generalizing the
> conditions under which a pair of functions on the line may be said to
> be Fourier transforms of each other, in a way which is independent of
> any special methods of summation and more inclusive than the usual
> $L_p$ theory. Our approach builds on earlier work by I.E. Segal, who
> suggested the definition "a measurable function $f$ has a generalized
> Fourier transform if the operation of convolution by $f$ in $L_2$ has
> a normal extension."

Kunze's definition of the generalized Fourier transform is contained in the following:

<IMG SRC="https://www.lorentz.leidenuniv.nl/beenakker/MO/Fourier_transform.png">

For an extension of this approach from the real line to compact Abelian groups, see K.I. Gross, <A HREF="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.99.4743&rep=rep1&type=pdf">Generalized Fourier Transforms of Distributions.</A>