PlanetMath: Fourier coefficients

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Fourier coefficients

(Definition)

Let $\mathbb{T}^n=\mathbb{R}^n/(2\pi\mathbb{Z})^n$ be the -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 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 with respect to the given basis. In particular, the Fourier series for converges to in the 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 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

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

The trigonometric series

$\displaystyle a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$

is called the trigonometric series of the function

, or Fourier series of the function

"Fourier coefficients" is owned by mathcam. [ full author list (2) | owner history (3) ]

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Also defines:

Fourier series, trigonometric series

Attachments:: example of Fourier series (Example) by alozano
common Fourier series (Example) by stevecheng
Fourier sine and cosine series (Topic) by pahio
Fourier series in complex form and Fourier integral (Derivation) by pahio

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Cross-references: Riemann integral, place, Lebesgue integral, trigonometric functions, function, Riemann integrable, norm, converges, basis, numbers, expand, orthonormal basis, torus
There are 32 references to this entry.

This is version 16 of Fourier coefficients, born on 2003-09-10, modified 2008-04-28.
Object id is 4716, canonical name is FourierCoefficients.
Accessed 18406 times total.

Classification:

AMS MSC:

11F30 (Number theory :: Discontinuous groups and automorphic forms :: Fourier coefficients of automorphic forms)

Pending Errata and Addenda

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