# Characterizing a functional that takes convolution to addition

Let $H:L^2[0,1]\rightarrow \mathbb{R}$ satisfy $$H(f*g)=H(f)+H(g).$$ Question:Is there a characterization of all such functionals $H$?

Related questions:Can it be extended to measures? If so, is it enough to know that the equation holds for discrete measures?

PS: I earlier asked this related question

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I gather here that by "functional" you don't mean "linear", right? Do you imply on the other hand a continuity assumption? – Loïc Teyssier Oct 16 '13 at 19:10
referring to this earlier related question you mention, isn't $H(f)$ just the logarithm of the (generalized) Fourier transform of $f$? – Carlo Beenakker Oct 16 '13 at 20:41
@LoïcTeyssier I struggled a bit trying to decide what assumptions to add, but since the question asks for a characterization, what extra assumptions are needed---if any---is part of the question. I hope this doesn't make the question too vague; I would accept an answer that assumes linearity. Do you think I should reword the question? – Henrique de Oliveira Oct 16 '13 at 22:15
@CarloBeenakker The Fourier transform has an image on complex-valued functions, so I believe the two are somewhat different. Besides, regarding the previous question, I'm still not sure if the Fourier transform is uniquely defined by that property. – Henrique de Oliveira Oct 16 '13 at 22:15
Alesker, Artstein, Faifman and Milman studied closely related questions in projecteuclid.org/… (see in particular their theorem 4). This applies to your map after exponentiation; they note that the real version of their result is in S. Alesker, S. Artstein-Avidan and V. Milman, A characterization of the Fourier transform and related topics, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 11–26. – ofer zeitouni Oct 17 '13 at 3:21