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# uniqueness of Fourier expansion

If a real function $f$, Riemann integrable on the interval $[-\pi,\,\pi]$, may be expressed as sum of a trigonometric series

$\displaystyle f(x)=\frac{a_{0}}{2}\!+\!\sum_{{m=1}}^{\infty}(a_{m}\cos{mx}+b_{% m}\sin{mx})$ | (1) |

where the series $a_{1}\!+\!b_{1}\!+\!a_{2}\!+\!b_{2}\!+\!a_{3}\!+\!b_{3}\!+\ldots$ of the coefficients converges absolutely, then the series (1) converges uniformly on the interval and can be integrated termwise. The same concerns apparently the series which are gotten by multiplying the equation (1) by $\cos{nx}$ and by $\sin{nx}$; the results of the integrations determine for the coefficients $a_{n}$ and $b_{n}$ the unique values

$\displaystyle a_{n}$ | $\displaystyle=\frac{1}{\pi}\!\int_{{-\pi}}^{{\pi}}f(x)\cos{nx}\,dx,$ | ||

$\displaystyle b_{n}$ | $\displaystyle=\frac{1}{\pi}\!\int_{{-\pi}}^{{\pi}}f(x)\sin{nx}\,dx$ |

for any $n$. So the Fourier series of $f$ is unique.

As a consequence, we can infer that the well-known goniometric formula

$\sin^{2}{x}=\frac{1-\cos{2x}}{2}$ |

presents the Fourier series expansion of the even function $\sin^{2}{x}$.

## Mathematics Subject Classification

42A20*no label found*42A16

*no label found*26A06

*no label found*

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