# Fourier coefficients

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

We can expand $f$ as a Fourier series

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

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 $\mathbb{R}$. Then the numbers

${a}_{0}$ | $={\displaystyle \frac{1}{2\pi}}{\displaystyle {\int}_{-\pi}^{\pi}}f(x)\mathit{d}x,$ | ||

${a}_{n}$ | $={\displaystyle \frac{1}{\pi}}{\displaystyle {\int}_{-\pi}^{\pi}}f(x)\mathrm{cos}(nx)\mathit{d}x,$ | ||

${b}_{n}$ | $={\displaystyle \frac{1}{\pi}}{\displaystyle {\int}_{-\pi}^{\pi}}f(x)\mathrm{sin}(nx)\mathit{d}x.$ |

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

$${a}_{0}+\sum _{n=1}^{\mathrm{\infty}}({a}_{n}\mathrm{cos}(nx)+{b}_{n}\mathrm{sin}(nx))$$ |

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

Title | Fourier coefficients |
---|---|

Canonical name | FourierCoefficients |

Date of creation | 2013-03-22 13:57:07 |

Last modified on | 2013-03-22 13:57:07 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 19 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11F30 |

Related topic | GeneralizedRiemannLebesgueLemma |

Related topic | FourierSeriesOfFunctionOfBoundedVariation |

Related topic | DirichletConditions |

Defines | Fourier series |

Defines | trigonometric series |