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1、FOURIER TRANSFORMSNOTED BY HUI MADefinition 0.1. Loosely speaking, for a given function f : R C the functionF() (for R) is defined by(1)F() =Zf(x)eixdx,provided the integral exists. The function F() is called the Fourier transform of f(x). Correspondingly,(2)f(x) =1 2ZF()eixdxis called the inverse F
2、ourier transform of F().Table 1. Properties of Fourier TransformsOperationf(x)F()Linearityaf(x) + bg(x)aF() + bG()Conjugationf(x)F()SymmetryF(x)2f()Scalingf(ax)1 |a|F( a)Differentiationf0(x)iF()Differentiationdnf dxn(i)nF()Differentiationixf(x)F0()Differentiation(ix)nf(x)F(n)()IntegrationRxf(t)dtF()
3、 i+ F(0)()Shiftf(x a)eiaF()Shifteiaxf(x),a RF( a)Convolutionf(x) g(x)F()G()Convolutionf(x)g(x)1 2(F G)()Date: November 3, 2004.12NOTED BY HUI MATable 2. Table of Fourier Transformsf(x) =1 2RF()eixdF() =Rf(x)eixdxDelta function(x)1Constant12()(x a),a Reiaeiax,a R2( a)cosine functioncos(ax) =eiax+eiax
4、 2,a R( a) + ( + a)sine functionsin(ax) =eiaxeiax 2i,a Ri( a) ( + a)Heaviside functionH(x) =1,if x 0;0,otherwise.() +1 iSign functionSign(x) = H(x) H(x) = 1,if x 0;0,if x = 0;1,otherwise.2 iPulse functionpa(x) =1,if |x| a 2; 0,otherwise.2sin(a/2) Gaussianeax2,a 0pae2 4aExponentialea|x|,a 02a a2+2Ref
5、erences1 R.J. Beerends, H.G. ter Morsche, J.C. van den Berg and E.M. van de Vrie, Fourier and Laplace transforms, Cambridge, 2003. (O174.22 FF77)2 Walter A. Strauss, Partial diffential equations, An introduction, John Wiley & Sons, Inc. 1992. (O175.2 FS 91)Department of Mathematical Sciences, Tsinghua University, Beijing 1000871, P.R. China
