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 FourierSeriesOfFunctionOfBoundedVariation 2013-03-22 17:58:00 2013-03-22 17:58:00 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 42A16 msc 42A20 msc 26A45 DirichletConditions FourierCoefficients