I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and exponential functions. For example: $$D = \left[\begin{array}{rr}0&1\\-1&0\end{array}\right] , \text{if} \sin = \left[\begin{array}{r}1\\0\end{array}\right] \text{and} \cos = \left[\begin{array}{r}0\\1\end{array}\right]$$ Another famous set of functions which have nice and simple such matrices are the polynomials.
Now to my question: are there sets of functions for which we can create same kind of matrices for function composition?
For instance if $p(x)$ and $q(x)$ polynomials, to find or systematically build a matrix such that $D_p q$ represents $p(q)$ as a vector.
Carleman matrices, defined as $$ M[f]_{jk} = \frac{1}{k!}\left[\frac{d^k}{dx^k} (f(x))^j\right]_{x = 0}, $$ convert function composition to matrix multiplication. The functions are required to be analytic at $0$ (for the usual definition) or at some $x_0$ (if derivatives are taken at $x_0$ instead of $0$ in the above definition). They satisfy $M[f \circ g] = M[f] M[g]$. Similar concepts are Bell matrices and Jabotinsky matrices.