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I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity: $$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$ Example of such ...

Let $$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$ where $y\in(0,\infty)$, $u\in(0,1)$, and $$f(t):=t+\pi (1-t) t \cot (\pi t).$$ Here are the graphs of $f$ (black), ...