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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 functions are given in this question: Connection between cyclic group and exponential function

and a possible generalization to arbitrary finite groups was given in this answer: https://mathoverflow.net/a/412397/165920

However in the answer only consequences of the convolution identity are shown but it is not shown how to construct such functions.

So my question is: Given the symmetric group $G = S_3$. Is it possible to construct functions $t_g$ which satisfy the convolution identity above?

I have added the tag for odes since the functions given as examples satisfy:

$$t^{'''}_{k}(x) = t_{k}(x), k=0,1,2$$

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    $\begingroup$ What is the domain and range of these functions? $\endgroup$
    – Wojowu
    Commented Dec 23, 2021 at 19:29
  • $\begingroup$ @Wojowu: $\mathbb{C}$ $\endgroup$
    – mathoverflowUser
    Commented Dec 23, 2021 at 19:30
  • $\begingroup$ This question has been answered here: mathoverflow.net/a/412397/165920 $\endgroup$
    – mathoverflowUser
    Commented Dec 26, 2021 at 17:14

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