PlanetMath: Fourier series of function of bounded variation

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Fourier series of function of bounded variation

(Theorem)

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

$\begin{align*}\begin{cases}a_n &= \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos{nx}\,dx\\ b_n &= \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin{nx}\,dx \end{cases}\end{align*}$

(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 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

$\begin{align*}\begin{cases}a_n &= \frac{1}{p}\int_{-p}^p f(t)\cos\frac{n\pi t}{p... ... &= \frac{1}{p}\int_{-p}^p f(t)\sin\frac{n\pi t}{p}\,dt. \end{cases}\end{align*}$

(4)

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

"Fourier series of function of bounded variation" is owned by pahio.

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See Also: Dirichlet conditions, Fourier coefficients

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Cross-references: limits, variable, equations, function, period, periodic function, represents, real, terms, periodicity, right-sided limit, arithmetic mean, interior points, series, sum, point, converges, Fourier series, interval, bounded variation, real function

This is version 4 of Fourier series of function of bounded variation, born on 2008-04-03, modified 2008-04-04.
Object id is 10472, canonical name is FourierSeriesOfFunctionOfBoundedVariation.
Accessed 831 times total.

Classification:

AMS MSC:	26A45 (Real functions :: Functions of one variable :: Functions of bounded variation, generalizations)
	42A16 (Fourier analysis :: Fourier analysis in one variable :: Fourier coefficients, Fourier series of functions with special properties, special Fourier series)
	42A20 (Fourier analysis :: Fourier analysis in one variable :: Convergence and absolute convergence of Fourier and trigonometric series)

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