There’s some confusion concerning representations of integration in the reciprocal spaces of the Fourier and Laplace transforms in some entries of MathOverflow that arises from not distinguishing among integration operators with different limits of integration (combined with some handwaving about group characters). Misassociation, or conflation, of distinct integration ops has been a source of historical confusion (cf. [Threefold Interpretation of Fractional Derivatives][1] by R. Hilfer as well as the comments in [MO question][2]). The convolution theorems provide a way to view differing integrations to effectively translate them into relatively simple factors in the reciprocal spaces.

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