Integral transform Laguerre function

Question

Given the following integral transform

$$ g(m)= \int_{0}^{\infty}dxe^{-x}f(x)L_{m}(x), $$

then how could we obtain $ f(x) $ from $ g(m) $ ??

I have thought that for a continuum '$m$' like in our integral transform we would have

$$ \int_{0}^{\infty}dxe^{-x}L_{n}(x)L_{m}(x)= \delta (n-m), $$ then how could we obtain $ f(x) $ from $ g(m) $ ?? For example, $$ f(x)=\int_{0}^{\infty}dmg(m)L_{m}(x)$$

Here $ L_{n}(x)= \frac{e^{x}}{\Gamma (n+1)}D^{m}(e^{-x}x^{n})$.

Answer 1

Here is how you get $f(x)$

$$ f(x)=\sum_{m=0}^{\infty} g(m)L_m(x). $$