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# 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*

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## 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