Fourier coefficients

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

We can expand $f$ as a Fourier series

${\displaystyle \sum _{k\in {\mathbb{Z}}^n}}\widehat{f}(k){\varphi }_k,$

and we call the numbers $\widehat{f}(k)$ 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 $ℝ$. Then the numbers

$a_0$$={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)𝑑x,$

$a_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos (nx)𝑑x,$

$b_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin (nx)𝑑x.$

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

$$a_0+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (nx)+b_n\sin (nx))$$

is called the trigonometric series of the function $f$, or Fourier series of the function $f.$