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单词 FourierSeriesOfFunctionOfBoundedVariation
释义

Fourier series of function of bounded variation


If the real function f is of bounded variationMathworldPlanetmath on the interval  [-π,+π],  then its Fourier series expansion

a02+n=1(ancosnx+bnsinnx)(1)

with the coefficients (http://planetmath.org/FourierCoefficients)

{an=1π-ππf(x)cosnxdxbn=1π-ππf(x)sinnxdx(2)

converges at every point of the interval. The sum of the series is at the interior points x equal to the arithmetic meanMathworldPlanetmath of the left-sided (http://planetmath.org/OneSidedLimit) and the right-sided limit of f at x and at the end-points of the interval equal to 12(limx-π+f(x)+limx+π-f(x)).

Remark 1.  Because of the periodicity of the terms of the terms, the expansion (1) converges for all real values of x and it represents a periodic functionMathworldPlanetmath with the period 2π.

Remark 2.  If the functionMathworldPlanetmath f is of bounded variation, instead of  [-π,+π],  on the interval [-p,+p]  the equations (1) and (2) may be converted via the change of variable x:=ptπ  to

a02+n=1(ancosnπtp+bnsinnπtp)(3)

and

{an=1p-ppf(t)cosnπtpdtbn=1p-ppf(t)sinnπtpdt.(4)

Correspondingly, the sum of (3) at the points of  [-p,+p]  is expressed via the left-sided and righr-sided limits of f(t).

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更新时间:2025/5/4 10:08:48