One of the main properties of the Laplace transform is given by the convolution theorem.
$$\mathcal{L}(f\*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$
**Question:** Is there a full characterization of the Laplace transform based on this property? I have in mind a theorem that reads: "Let $\mathcal{N}$ be an operator with the convolution property (given above). If $\mathcal{N}$ also satisfies properties A,B,C then $\mathcal{N}=\mathcal{L}$."



Characterization of the Laplace Transform
This operational approach to a definition of Fourier/Laplace transforms has been developed by R.A. Kunze, [An operator theoretic approach to generalized Fourier transforms](http://www.jstor.org/stable/1970090).



> 
> 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:


![](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, [Generalized Fourier Transforms of Distributions.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.99.4743&rep=rep1&type=pdf)


