I'm familiar with the slogan "the Fourier transform decomposes signals into frequencies", but I've still somehow managed to get by without really understanding what the Fourier transform is or why it is useful.
My understanding is that it is a sort of change of basis on the space of functions. Is this accurate? What exactly is the basis we are changing to, and which basis are we changing from?
I'm also familiar with the characteristic function from probability theory, or the Fourier transform of the pdf of a random variable, and the fact that it determines the distribution of a random variable. Other than making a couple proofs a bit nicer, what exactly is the significance of the characteristic function?
Consider a function $h$. This function can be made by adding functions of the form $\exp(i \omega)$ together. You might ask, “How much of each $\omega$ do I need?” I could tell you this answer by giving you a complex number for each $\omega$; that is, I could give you a new function of $\omega$. That new function is the Fourier transform of $h$ and the different $\omega$s are called frequencies.