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# 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

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

## Mathematics Subject Classification

42A16*no label found*42A20

*no label found*26A45

*no label found*

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