this should be simple
A polynomial could be defined as \begin{equation} P_n (x) = \sum_{i=1}^{n} a_i x^{i-1} \end{equation}
Would the infinite-dimensional version of that \begin{equation} F_l (x) = \int_{0}^{l} a(y) x^y dy \end{equation} already have some name that everybody else than me already knows?