Geodesic
A function is determined by the norms of its convolutions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 179-181 
A. I. Plotkin. A function is determined by the norms of its convolutions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 179-181. http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a16/
@article{ZNSL_1974_47_a16,
     author = {A. I. Plotkin},
     title = {A~function is determined by the norms of its convolutions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {179--181},
     year = {1974},
     volume = {47},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a16/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $G$ be a compact abelian group and $f$, $g\in L^p(G)$, $p$ is not even. Let $\varphi_a$ denote the $a$-shift of the function $\varphi$. It is proved that $$ \bigl\|\sum\alpha_if_{a_i}\bigr\|_{L^p}=\bigl\|\sum\alpha_ig_{a_i}\bigr\|_{L^p}, $$ for all $\alpha_1,\dots,\alpha_n\in\mathbb R^1$ and $a_1,\dots,a_n\in G$ then there exist $b\in G$ and $\alpha\in\mathbb R^1$, $|\alpha|=1$, such that $f=\alpha g_b$.