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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?

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    You may have a look at the Mellin transform and its inverse: http://en.wikipedia.org/wiki/Mellin_transform2011-05-05

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With the substitution $x=e^{i\omega}$ is is known as Fourier-transform.

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    Yes, like the polynomial would be discrete Fourier transfom or spectral decomposition.2011-05-05