# What is Fourier Space

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: $\frac{d}{dt}\hat u_k=(k^2-k^4)\hat u_k+\sum_{k'}k'\hat u_{k'}\hat u_{k-k'}$

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