Editing Fourier series (section)

=== Fourier series on a square ===
We can also define the Fourier series for functions of two variables <math>x</math> and <math>y</math> in the square <math>[-\pi,\pi]\times[-\pi,\pi]</math>:
<math display="block">\begin{align}
f(x,y) & = \sum_{j,k \in \Z} c_{j,k}e^{ijx}e^{iky},\\[5pt]
c_{j,k} & = \frac{1}{4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.
\end{align}</math>

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in [[image compression]]. In particular, the [[JPEG]] image compression standard uses the two-dimensional [[discrete cosine transform]], a discrete form of the [[Sine and cosine transforms|Fourier cosine transform]], which uses only cosine as the basis function.

For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.<ref>[https://www.youtube.com/watch?v=V7l9Im9zneg Vanishing of Half the Fourier Coefficients in Staggered Arrays]</ref>