傅立叶展开与傅立叶变换
1.傅立叶展开
对于周期为的周期
函
关于工期滞后的函关于工程严重滞后的函关于工程进度滞后的回复函关于征求同志党风廉政意见的函关于征求廉洁自律情况的复函
数,可展开为 ,,fx2,
,a0 ,,,,fx,,acosnx,bsinnxnn,2n1,
1函数系,,,,,……在上满足正交性 ,,cosxsinxcos2xsin2x,,,,
2
,,1111 dx,1sinmxcosnxdx,0,,,,,,,,22
,,1122 sinnx,0cosnx,0,,,,,,,,
,,11 m,n,sinmxsinnx,0m,n,cosmxcosnx,0,,,,,,,,
,1则 ,,fxcosnxdx,an,0n,,,,
,1 ,,fxsinnxdx,bn,1n,,,,
对于任意周期为的周期函数,可展开为 ,,fxT,2l
,ann,,,,0 ,,fx,,acosx,bsinx,,nn,2ll,,n1,
1,,2,2,函数系,,,,,……在上满足正交性 cosxsinxxx,,,l,lcossinllll2
,,1111 dx,1sinmxcosnxdx,0,,,,,,,,22
,,1122 sinnx,0cosnx,0,,,,,,,,
,,11 m,n,sinmxsinnx,0m,n,cosmxcosnx,0,,,,,,,,
l1n,则 ,,fxcosxdx,an,0n,,lll
l1n, ,,fxsinxdx,bn,1n,,lll
2.傅立叶变换
对于无周期的函数,,,其定义在,,上,可看作为的周期函数取fx,,,,T,2l
的极限。 l,,
,ann,,,,0,, fx,,acosx,bsinx,,nn,2ll,,n1,
ll1n1n,, ,,,,fxcosxdx,afxsinxdx,bnn,,,,llllll,ll11ntx,,,, ?,,,,,,fxftdtftcosdt,,,,,ll,,2llln1,
l1当时, ,,ftdt,0l,,,,l2l
n,,再取, ,,,,,nnll
,l1则 ,,,,,,,ftcos,t,xdtfx,n,l,ln1,,l1,, ,,,,,ftcos,t,xdt,,,,nn,,l,,,,n1,
,,1 ,,,,,d,ftcos,t,xdt,,0,,,
定义傅氏积分:
,,1 ,,,,,,fx,d,ftcos,t,xdt,,0,,,
傅立叶定理:若函数在任意有限区间上满足狄利克雷条件,且在区间,,fx
内绝对可积,则的傅氏积分在上处处收敛,且有 ,,,,,,fx,,,,,,,,
,,100fx,,,,,,fx,cos ,,,,,,dftt,xdt,,,0,,2,
此式称为傅氏积分公式。
利用欧拉公式,此公式可写为复数形式:
,,1,0,,0,,,,fxfxi,,,tx,, ,,,dftedt,,,,,,22,
当在处连续时,则在处的傅氏积分就等于。 ,,,,,,fxxfxxfx
,,1i,,,tx, ,,,,fx,d,ftedt,,,,,,2,
,,1i,ti,x,,, ,,,ftedted,,,,,,,,,,,2,,i,x令 ,,,,F,,fxedx,,,
,1i,x,则 ,,,,fx,F,ed,,,,2,
被称为的傅氏变换,被称为的傅氏逆变换。,,,,,,,,F,fxfxF,