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

$\displaystyle\frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos{nx}+b_{n}\sin{nx})$

(1)

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

$\displaystyle\begin{cases}a_{n}&=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos{nx}\,% dx\\ b_{n}&=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin{nx}\,dx\end{cases}$

(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 $\displaystyle\frac{1}{2}\left(\lim_{x\to-\pi+}f(x)+\lim_{x\to+\pi-}f(x)\right)$ .

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 $\displaystyle x:=\frac{pt}{\pi}$ to

$\displaystyle\frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos\frac{n\pi t}{p}+b_{% n}\sin\frac{n\pi t}{p})$

(3)

and

$\displaystyle\begin{cases}a_{n}&=\frac{1}{p}\int_{-p}^{p}f(t)\cos\frac{n\pi t}% {p}\,dt\\ b_{n}&=\frac{1}{p}\int_{-p}^{p}f(t)\sin\frac{n\pi t}{p}\,dt.\end{cases}$

(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