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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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Your function is not really a generalization of a polynomial since the exponent is not a natural number; a better generalization might be $\sum_{n=0}^\infty a_nx^n$ which is the very-well known and important concept of power series. – Gadi A May 5 '11 at 9:04
Well it is not polynomial as it is not a polynomial that's true. But I think there is certain similarity. – etorri May 5 '11 at 9:31
Indeed; I just wanted to point out another, maybe more "in the spirit of polynomials" possible generalization. – Gadi A May 5 '11 at 10:23
You may have a look at the Mellin transform and its inverse: – Dirk May 5 '11 at 11:42

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

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Or the substitution $x = e^{-s}$, with $l = \infty$ gives the Laplace transform... – Sangchul Lee May 5 '11 at 9:34
Yes, like the polynomial would be discrete Fourier transfom or spectral decomposition. – etorri May 5 '11 at 9:38

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