Fourier series of function of bounded variation

If the real function $f$ is of bounded variation on the interval  $[-\pi ,+\pi ]$,  then its Fourier series expansion

${\displaystyle \frac{a_0}{2}}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(a_n\cos nx+b_n\sin nx)$ (1)

with the coefficients (http://planetmath.org/FourierCoefficients)

(2)

converges at every point of the interval. The sum of the series is at the interior points $x$ equal to the arithmetic mean of the left-sided (http://planetmath.org/OneSidedLimit) and the right-sided limit of $f$ at $x$ and at the end-points of the interval equal to  ${\displaystyle \frac{1}{2}}\left(\underset{x\to -\pi +}{lim}f(x)+\underset{x\to +\pi -}{lim}f(x)\right)$.

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 function with the period $2\pi$.

Remark 2.  If the function $f$ is of bounded variation, instead of  $[-\pi ,+\pi ]$,  on the interval  $[-p,+p]$  the equations (1) and (2) may be converted via the change of variable  $x:={\displaystyle \frac{pt}{\pi }}$  to

${\displaystyle \frac{a_0}{2}}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(a_n\cos {\displaystyle \frac{n\pi t}{p}}+b_n\sin {\displaystyle \frac{n\pi t}{p}})$ (3)

and

(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