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[parent] uniqueness of Fourier expansion (Result)

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 $\sin^2{x}$ .



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See Also: Fourier sine and cosine series, minimality property of Fourier coefficients, determination of Fourier coefficients, complex sine and cosine, uniqueness of digital representation

Other names:  uniqueness of Fourier series

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Cross-references: even function, goniometric formula, consequence, Fourier series, equation, converges uniformly, converges absolutely, coefficients, series, trigonometric series, sum, interval, Riemann integrable, real function

This is version 2 of uniqueness of Fourier expansion, born on 2008-09-09, modified 2008-09-09.
Object id is 11012, canonical name is UniquenessOfFourierExpansion.
Accessed 1023 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
 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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