I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually Taylor series terms) and built some of my own. I noticed that for any finite size of such matrix, it can only represent up to a finite size polynomial.
My question is two-fold:
Is there any other basis of functions for some other type of matrix which can represent function concatenation - and in general are there any extra suitable functions which have nice properties with concatenation (except polynomials)?
To expand Carleman matrices to be able to represent multivariable functions, a hypothesis of mine is that it would be possible to treat as a multivariable taylor polynomial. Does someone know if this has been done?