# 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 (http://planetmath.org/SumFunctionOfSeries).  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 of the even function $\sin^{2}{x}$.

 Title uniqueness of Fourier expansion Canonical name UniquenessOfFourierExpansion Date of creation 2013-03-22 18:22:16 Last modified on 2013-03-22 18:22:16 Owner pahio (2872) Last modified by pahio (2872) Numerical id 5 Author pahio (2872) Entry type Result Classification msc 42A20 Classification msc 42A16 Classification msc 26A06 Synonym uniqueness of Fourier series Related topic FourierSineAndCosineSeries Related topic MinimalityPropertyOfFourierCoefficients Related topic DeterminationOfFourierCoefficients Related topic ComplexSineAndCosine Related topic UniquenessOfDigitalRepresentation Related topic UniquenessOfLaurentExpansion