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 $f.$

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