Is there a way to double all frequencies? Let's imagine I have an input signal that looks like this:

$$
f(t)=\sin\Big(2\pi(2)t\Big)+\sin\Big(2\pi(3)t\Big)+\sin\Big(2\pi(5)t\Big)
$$



And I want an output signal that scales each frequency proportionally by some factor. Here, I scale everything by a factor of '2':



Is there a way to do this?  The frequency domains look like this:
That's typically referred to as "pitch shifting" and it's typically a combination of re-sampling and time stretch/compression. Re-sampling is straight forward but time stretch/compression is a bit more tricky and the best methods depends on the application requirements, the characteristics of the signal and the amount of stretch/compression you'll need. See http://blogs.zynaptiq.com/bernsee/time-pitch-overview/ and https://en.wikipedia.org/wiki/Audio_time-scale/pitch_modification to get started