I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this is.

Here's an example for context: For the pde $(1)$ $u_t+u_{xxxx}+u_{xx}+uu_x=0$, where $x\in [-L/2,L/2]$, the term $u_{xx}$ is responsible for instability at large scales and $u_{xxxx}$ provides dampening at small scales. This is readily apparent in Fourier space, where one may write $(1)$ with periodic boundary conditions as: $\frac{d}{dt}\hat u_k=(k^2-k^4)\hat u_k+\sum_{k'}k'\hat u_{k'}\hat u_{k-k'}$

  • 2
    $\begingroup$ Fourier space is just lingo for what a function looks like after a Fourier transform. $\endgroup$ – Cameron Williams Jun 2 '14 at 2:29
  • $\begingroup$ Ah, ok. Thanks very much! $\endgroup$ – user153582 Jun 2 '14 at 2:32

When a function $f$ is defined on $\mathbb R^n$, its Fourier transform $\hat f$ is also defined on $\mathbb R^n$. But it would be counterproductive to think of both $f$ and $\hat f$ as coinhabiting the same space. Forming expressions like $f+\hat f$ would be nonsensical. The functions are really defined on different spaces. To emphasize this distinction, one can call the domain of $f$ physical space, and the domain of $\hat f$ Fourier space.

When $n=1$, it makes physical sense to call the variable of $f$ time and the variable of $\hat f$ frequency. Following this, one can also say that $\hat f$ lives in the *frequency space_ (even if $n>1$), to emphasize the above distinction.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.