Fourier coefficients

Let 𝕋n=n/(2π)n be the n-dimensional torus, let {ϕk(x)}kn be an orthonormal basis for L2(𝕋n), and suppose that f(x)L2(𝕋n).

We can expand f as a Fourier series


and we call the numbers 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 L2 norm.

The most basic incarnation of this is finding the Fourier coefficients of a Riemann integrablePlanetmathPlanetmath functionMathworldPlanetmath with respect to the orthonormal basis given by the trigonometric functionsDlmfMathworldPlanetmath:

Let f be a Riemann integrable function from [-π,π] to . Then the numbers

a0 =12π-ππf(x)𝑑x,
an =1π-ππf(x)cos(nx)𝑑x,
bn =1π-ππf(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 integralMathworldPlanetmath in place of the Riemann integral. This is the usual setting for modern Fourier analysis.

The trigonometric series


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