Why fourier transform tell us energy of any frequency of f(t)

In signal processing, the energy of a continuous-time signal is defined as the square of its $L^2$ norm. Hence, the spectral energy of this signal is the square of the $L^2$ norm of this signal in the spectral domain, i.e. of its Fourer transform. By Parceval's theorem, these two energies are equal. But as a consequence, $|x(t)|^2$ is the energy density of the signal at the moment $t$, and $|F(x)(tau)|^2$ is the spectral energy density at the frequency $tau$.

This is in complete analogue with the discrete case: in your notation, $a_n^2 + b_n^2$ is equal (up to a multiplicative constant) to the square of the absolute value of $int f(t) e^{-i theta t} dt$ for $theta=n$ (for the real-valued $f$).

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