I have been confused by a notation (?) of Fourier transformation. When I was reading a paper, I saw a definition of Fourier transform and its inverse as
$(\mathcal{F}f)(k)=\int_{D}f(x)e^{-2i\pi<x,k>} {\rm{d}}x,\quad (\mathcal{F}f)(x)=\int_{D}f(k)e^{2i\pi<x,k>} {\rm{d}}k$.
I have two questions with these equations.
- Isn't $k$ frequency, like $\omega$? Fourier transform maps a function in spatial domain to frequency domain, right? However, it seems that $k$ here is a integer.
- What is $<x,k>$? Is this just a multiplication? What is implied by putting them in <>?
I might be asking silly questions due to my background, but it would be super helpful if you could give me some hints to solve my questions. Thank you so much.