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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.

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:

For an extension of this approach from the real line to compact Abelian groups, see K.I. Gross, Generalized Fourier Transforms of Distributions.