Fourier series of function of bounded variation

If the real function f is of bounded variationMathworldPlanetmath on the interval[-π,+π],  then its Fourier series expansion

a02+n=1(ancosnx+bnsinnx) (1)

with the coefficients (

{an=1π-ππf(x)cosnxdxbn=1π-ππf(x)sinnxdx (2)

converges at every point of the interval. The sum of the series is at the interior points x equal to the arithmetic meanMathworldPlanetmath of the left-sided ( and the right-sided limit of f at x and at the end-points of the interval equal to  12(limx-π+f(x)+limx+π-f(x)).

Remark 1.  Because of the periodicity of the terms of the terms, the expansion (1) converges for all real values of x and it represents a periodic functionMathworldPlanetmath with the period 2π.

Remark 2.  If the functionMathworldPlanetmath f is of bounded variation, instead of  [-π,+π],  on the interval  [-p,+p]  the equations (1) and (2) may be converted via the change of variable  x:=ptπ  to

a02+n=1(ancosnπtp+bnsinnπtp) (3)


{an=1p-ppf(t)cosnπtpdtbn=1p-ppf(t)sinnπtpdt. (4)

Correspondingly, the sum of (3) at the points of  [-p,+p]  is expressed via the left-sided and righr-sided limits of f(t).

Title Fourier series of function of bounded variation
Canonical name FourierSeriesOfFunctionOfBoundedVariation
Date of creation 2013-03-22 17:58:00
Last modified on 2013-03-22 17:58:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 42A16
Classification msc 42A20
Classification msc 26A45
Related topic DirichletConditions
Related topic FourierCoefficients