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

$f(x)={\displaystyle \frac{a_0}{2}}+{\displaystyle \sum _{m=1}^{\mathrm{\infty }}}(a_m\cos mx+b_m\sin mx)$ (1)

where the series $a_1+b_1+a_2+b_2+a_3+b_3+\mathrm{\dots }$ 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

$a_n$ $={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos nxdx,$

$b_n$ $={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin nxdx$

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