Assume I have some sampling of a function $f(t)$ at points $t$:
$$f(t_k) = d_k, \forall k \in \{1,\cdots,n\}$$
Assume we have vectors $\bf v_k$ which can be functions of $x_k$ and $d_k$, can we find a group so that it has a matrix representation in such a way that there exists a generating element which has representation matrix $\bf M$, so that:
$${\bf Mv_{k}} = {\bf v_{k+1}}, \forall k$$
Or rather, for which families of functions is this powerful enough to capture what we want?
( I am not interested in silly trivial solutions which give us no information about $f$, like for example "just let $\bf M$ and $\bf v_k$ zero, lol". )
A simple example of function I am interested in is scalar case: $$x_k = \{0,1,\cdots,n\}\\ f(x) = c_2\cdot {c_1}^x$$, where we get away with such a small/simple space as $\mathbb R^1$ $${\bf M}=c_1\\ {\bf v_k}=d_k$$