## You are here

HomeFourier coefficients

## Primary tabs

# Fourier coefficients

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

$\displaystyle\sum_{{k\in\mathbb{Z}^{n}}}\hat{f}(k)\phi_{k},$ |

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

$\displaystyle a_{0}$ | $\displaystyle=\frac{1}{2\pi}\int_{{-\pi}}^{{\pi}}f(x)dx,$ | ||

$\displaystyle a_{n}$ | $\displaystyle=\frac{1}{\pi}\int_{{-\pi}}^{{\pi}}f(x)\cos(nx)dx,$ | ||

$\displaystyle b_{n}$ | $\displaystyle=\frac{1}{\pi}\int_{{-\pi}}^{{\pi}}f(x)\sin(nx)dx.$ |

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.

$a_{0}+\sum_{{n=1}}^{{\infty}}(a_{n}\cos(nx)+b_{n}\sin(nx))$ |

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

## Mathematics Subject Classification

11F30*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

## Info

## Attached Articles

common Fourier series by stevecheng

Fourier sine and cosine series by pahio

Fourier series in complex form and Fourier integral by pahio

minimality property of Fourier coefficients by pahio

uniqueness of Fourier expansion by pahio

determination of Fourier coefficients by pahio