uniqueness of Fourier expansion

If a real function f, Riemann integrablePlanetmathPlanetmath on the interval[-π,π],  may be expressed as sum of a trigonometric series

f(x)=a02+m=1(amcosmx+bmsinmx) (1)

where the series a1+b1+a2+b2+a3+b3+ 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 cosnx and by sinnx;  the results of the integrations determine for the coefficients an and bn the unique values

an =1π-ππf(x)cosnxdx,
bn =1π-ππf(x)sinnxdx

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

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


presents the Fourier series of the even functionMathworldPlanetmath sin2x.

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