Let $\mathbb{T}^n=\mathbb{R}^n/(2\pi\mathbb{Z})^n$ be the $n$ -dimensional torus, let $\{\phi_k(x)\}_{k\in\mathbb{Z}^n}$ be an orthonormal basis for $L^2(\mathbb{T}^n)$ , and suppose that $f(x)\in L^2(\mathbb{T}^n)$ .

We can expand $f$ as a Fourier series

\begin{align*} \sum_{k\in\mathbb{Z}^n}\hat{f}(k)\phi_k, \end{align*} and we call the numbers $\hat{f}(k)$ the Fourier coefficients of $f$ with respect to the given basis. In particular, the Fourier series for $f$ converges to $f$ in the $L^2$ norm.

The most basic incarnation of this is finding the Fourier coefficients of a Riemann integrable function with respect to the orthonormal basis given by the trigonometric functions:

Let $f$ be a Riemann integrable function from $[-\pi,\pi]$ to $\mathbb{R}$ . Then the numbers \begin{align*} a_0 &= \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx,\\ a_n &= \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx,\\ b_n &= \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx. \end{align*}are called the Fourier coefficients of the function $f.$

The above can be repeated for a Lebesgue-integrable function $f$ if we use the Lebesgue integral in place of the Riemann integral. This is the usual setting for modern Fourier analysis.

The trigonometric series $$ a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$$ is called the trigonometric series of the function $f$ , or Fourier series of the function $f.$