TITLE: Real-valued 2D Fourier series?
QUESTION [17 upvotes]: For a (well-behaved) one-dimensional function $f: [-\pi, \pi] \rightarrow \mathbb{R}$, we can use the Fourier series expansion to write
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n\sin(nx) \right)$$
For a function of two variables, Wikipedia lists the formula
$$f(x,y) = \sum_{j,k \in \mathbb{Z}} c_{j,k} e^{ijx}e^{iky}$$
In this formula, $f$ is complex-valued. Is there a similar series representation for real-valued functions of two variables?

REPLY [20 votes]: The full real-valued 2D Fourier series is:
$$
\begin{align}
f(x, y) & = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\alpha_{n,m}cos\left(\frac{2\pi n x}{\lambda_x}\right)cos\left(\frac{2\pi m y}{\lambda_y}\right) \\
& + \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\beta_{n,m}cos\left(\frac{2\pi n x}{\lambda_x}\right)sin\left(\frac{2\pi m y}{\lambda_y}\right) \\
& + \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\gamma_{n,m}sin\left(\frac{2\pi n x}{\lambda_x}\right)cos\left(\frac{2\pi m y}{\lambda_y}\right) \\
& + \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\delta_{n,m}sin\left(\frac{2\pi n x}{\lambda_x}\right)sin\left(\frac{2\pi m y}{\lambda_y}\right) \\
\end{align}
$$
The coefficients are found with:
$$
\alpha_{n,m} = \frac{\kappa}{\lambda_x \lambda_y}\int_{y_0}^{y_0+\lambda_y}\int_{x_0}^{x_0+\lambda_x}f(x,y)cos\left(\frac{2\pi n x}{\lambda_x}\right)cos\left(\frac{2\pi m y}{\lambda_y}\right)dx dy \\
\beta_{n,m} = \frac{\kappa}{\lambda_x \lambda_y}\int_{y_0}^{y_0+\lambda_y}\int_{x_0}^{x_0+\lambda_x}f(x,y)cos\left(\frac{2\pi n x}{\lambda_x}\right)sin\left(\frac{2\pi m y}{\lambda_y}\right)dx dy \\
\gamma_{n,m} = \frac{\kappa}{\lambda_x \lambda_y}\int_{y_0}^{y_0+\lambda_y}\int_{x_0}^{x_0+\lambda_x}f(x,y)sin\left(\frac{2\pi n x}{\lambda_x}\right)cos\left(\frac{2\pi m y}{\lambda_y}\right)dx dy \\
\delta_{n,m} = \frac{\kappa}{\lambda_x \lambda_y}\int_{y_0}^{y_0+\lambda_y}\int_{x_0}^{x_0+\lambda_x}f(x,y)sin\left(\frac{2\pi n x}{\lambda_x}\right)sin\left(\frac{2\pi m y}{\lambda_y}\right)dx dy
$$
$$
\begin{align}
\text{Where } \kappa & = 1 \text{ if } n = 0 \text{ and } m = 0 \\
& = 2 \text{ if } n = 0 \text{ or } m = 0\\
& = 4 \text{ if } n> 0 \text{ and } m > 0
\end{align}
$$
Example plot