Modulation and filtering
This is the core subject of linear modulation theory that's considered in analog communication systems. It's analysis is based on the modulation property of the Fourier trabsform.

Given a bandlimited baseband signal $x(t)$ such that its CTFT $X(\omega)$ is zero outside of the band  $-W &lt; \omega &lt; W$, its multiplication with the signal $\cos(\omega_0t)$ has the following spectrum:

$$ x(t) \cos(\omega_0 t) \longleftrightarrow \frac{1}{2\pi} X(\omega) \star \mathcal{F} \{ cos(\omega_0 t) \} $$ where $\star$ stands for convolution.

The Fourier transform of $\cos(\omega_0 t)$ is
$$  \cos(\omega_0 t) \longleftrightarrow \pi \delta(w-\omega_0) + \pi \delta(w+\omega_0)$$

Using the convolution property $ f(x) \star \delta(x-a) = f(x-a)$ i.e., signal shifts to the impulse position, we deduce the spectrum of the modulated signal as:

$$ x(t) \cos(\omega_0 t) \longleftrightarrow \frac{1}{2\pi}  \left( \pi X(\omega-\omega_0) + \pi X(\omega+\omega_0) \right) = 0.5 X(\omega-\omega_0) + 0.5 X(\omega+\omega_0)$$

Now if you filter (convolve) this modulated signal with a (n ideal) BPF with a center frequency of $\omega_c = \omega_0$ and a bandwidth of at least $W_{BW} = 2 W$ then you would get the modulated signal spectrum alone. This can be shown by utilizing the filtering property of Fourier transform which says
$$ h(t) \star x(t) \longleftrightarrow H(\omega) X(\omega) $$

Now the defintion of the (ideal) BPF is such that its frequency response is:
$$
H(\omega) = 
\begin{cases}
1 &amp;, |\omega-\omega_0| &lt; W \\
0 &amp;, o.w. \\
\end{cases}
$$

Then the multiplication of $H(\omega)$ with $0.5 X(\omega-\omega_0) + 0.5 X(\omega+\omega_0)$ would yield:

$$ H(\omega) \left[ 0.5 X(\omega-\omega_0) + 0.5 X(\omega+\omega_0) \right] = 0.5 X(\omega-\omega_0) + 0.5 X(\omega+\omega_0) $$

Which means: 
$$ \mathcal{F} \{ \text{BPF} \{ x(t) \cos(\omega_0 t) \} \}= 0.5 X(\omega-\omega_0) + 0.5 X(\omega+\omega_0)$$  

In this example the whole spectrum is the modulated spectrum so you don't actually filter out anything, but in other situations the BPF would indeed perform some useful operations.