傅里葉變換||數(shù)學(xué)物理方法

2021-02-14 21:32 作者:湮滅的末影狐 0人讀過(guò) | 我要投稿

//其實(shí)傅里葉級(jí)數(shù)在3Blue1Brown的視頻里面已經(jīng)有比較詳細(xì)的介紹，但為了保證文集的完整性，這一篇筆記還是發(fā)表出來(lái)。

5.1 傅里葉級(jí)數(shù)

沒(méi)有周期函數(shù)： $f(x%2B2l)%3Df(x)$

取三角函數(shù)族：

$1%2C%5C%3B%5Ccos%5Cfrac%7B%5Cpi%20x%7D%7Bl%7D%2C%5C%3B%5Ccos%5Cfrac%7B2%5Cpi%20x%7D%7Bl%7D%2C...%2C%5Ccos%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%2C...%5C%5C%0A0%2C%5C%3B%5Csin%5Cfrac%7B%5Cpi%20x%7D%7Bl%7D%2C%5C%3B%5Csin%5Cfrac%7B2%5Cpi%20x%7D%7Bl%7D%2C...%2C%5Csin%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%2C...$

則該三角函數(shù)族正交：該函數(shù)族的任意兩不同函數(shù)在一周期的積分為0.

$%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Csin%20mx%20%5Ccos%20nx%20%5Cmathrm%20d%20x%3D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Ccos%20mx%20%5Ccos%20nx%20%5Cmathrm%20d%20x%5C%5C%3D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Csin%20mx%20%5Csin%20nx%20%5Cmathrm%20d%20x%3D0%5C%3B%5C%3B(m%2Cn%5Cin%20%5Cmathbb%20N%5E%2B%20%2C%20m%5Cneq%20n)$

將 $f(x)$ 展開(kāi)為級(jí)數(shù)：

$f(x)%3Da_0%2B%5Csum_%7Bk%3D1%7D%5E%5Cinfty(a_k%5Ccos%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%2Bb_k%5Csin%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D)$

則根據(jù)函數(shù)族的正交性，有

$%5Cint_%7B-l%7D%5El%20f(x)%20%5Ccos%20%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%20%5Cmathrm%20d%20x%3D%5Cint_%7B-l%7D%5El%20a_k%5Ccos%5E2%20%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%20%5Cmathrm%20d%20x%3Da_kl%20%5C%3B(k%5Cneq0)$

$%5Cint_%7B-l%7D%5El%20f(x)%20%5Csin%20%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%20%5Cmathrm%20d%20x%3D%5Cint_%7B-l%7D%5El%20b_k%5Csin%5E2%20%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%20%5Cmathrm%20d%20x%3Db_kl$

從而可以求出 $a_k%2Cb_k$ .?

$%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D%0Aa_%7Bk%7D%3D%5Cfrac%7B1%7D%7B%20l%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(%5Cxi)%20%5Ccos%20%5Cfrac%7Bk%20%5Cpi%20%5Cxi%7D%7Bl%7D%20%5Cmathrm%7B~d%7D%20%5Cxi%20%5C%5C%0Ab_%7Bk%7D%3D%5Cfrac%7B1%7D%7Bl%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(%5Cxi)%20%5Csin%20%5Cfrac%7Bk%20%5Cpi%20%5Cxi%7D%7Bl%7D%20%5Cmathrm%7B~d%7D%20%5Cxi%0A%5Cend%7Barray%7D%5Cright.$

特別地，

$a_0%3D%5Cfrac%7B%5Cint_%7B-l%7D%5Elf(x)%5Cmathrm%20d%20x%7D%7B2l%7D$

可以證明，這里的三角函數(shù)族是完備的：

$%5Cforall%20f(x)$ 連續(xù)， $n%5Crightarrow%5Cinfty$ ，

$%5Cint_%7B-l%7D%5E%7Bl%7D%5Bf(x)%5D%5E%7B2%7D%20%5Cmathrm%7B~d%7D%20x%3D%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%20a_%7Bk%7D%5E%7B2%7D%5Cleft%5B%5Ccos%20%5Cfrac%7Bk%20%5Cpi%20x%7D%7Bl%7D%5Cright%5D%5E%7B2%7D%2B%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D%20b_%7Bk%7D%5E%7B2%7D%5Cleft%5B%5Csin%20%5Cfrac%7Bk%20%5Cpi%20x%7D%7Bl%7D%5Cright%5D%5E%7B2%7D$

滿足上式，稱上述三角函數(shù)族完備，上式稱為完備性方程，稱級(jí)數(shù)平均收斂于 $f(x)$ .

狄里希利定理：若函數(shù)在每一周期內(nèi)除有限個(gè)第一類間斷點(diǎn)外處處連續(xù)，且只有有限個(gè)極值點(diǎn)，則前述傅里葉級(jí)數(shù)收斂，且在間斷點(diǎn)有

$a_0%2B%5Csum_%7Bk%3D1%7D%5E%5Cinfty(a_k%5Ccos%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%2Bb_k%5Csin%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D)%3D%5Cfrac12%5Bf(x%2B0)%2Bf(x-0)%5D$

什么收斂發(fā)散，嚴(yán)格處理起來(lái)真的好麻煩...作為物理人，我們只關(guān)心：能用就行...

對(duì)于奇函數(shù)，傅里葉級(jí)數(shù)只有正弦項(xiàng)；偶函數(shù)則只有余弦項(xiàng)。

傅里葉級(jí)數(shù)有復(fù)數(shù)形式。

$f(z)%3D%5Csum_%7Bk%3D-%5Cinfty%7D%5E%5Cinfty%20c_k%20e%5E%7B%5Cmathrm%20i%20%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%7D$

關(guān)于函數(shù)族的討論，與前面類似。本來(lái)復(fù)指數(shù)和三角函數(shù)的本質(zhì)就是一樣的。

在我的教材里面，求系數(shù)的公式是

$c_k%3D%5Cfrac%7B1%7D%7B2l%7D%5Cint_%7B-l%7D%5El%20f(%5Cxi)%5Be%5E%7B%5Cmathrm%20i%5Cfrac%7Bk%5Cpi%20%5Cxi%7D%7Bl%7D%7D%5D%5E*%5Cmathrm%20d%20%5Cxi$

但是我比較習(xí)慣的形式是

$c_k%3D%5Cfrac%7B1%7D%7B2l%7D%5Cint_%7B-l%7D%5El%20f(%5Cxi)e%5E%7B-%5Cmathrm%20i%5Cfrac%7Bk%5Cpi%20%5Cxi%7D%7Bl%7D%7D%5Cmathrm%20d%20%5Cxi$

本質(zhì)上是一樣的。

5.2 傅里葉積分與傅里葉變換

定義在 $%5Cmathbb%20R$ 的函數(shù)如果不是周期性的，就不能展開(kāi)為傅里葉級(jí)數(shù)，但可以考慮它是周期為 $2l$ 的函數(shù) $g(x)$ 令 $l%20%5Crightarrow%20%5Cinfty$ 的結(jié)果。

$g(x)%3Da_0%2B%5Csum_%7Bk%3D1%7D%5E%5Cinfty(a_k%5Ccos%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D%2Bb_k%5Csin%5Cfrac%7Bk%5Cpi%20x%7D%7Bl%7D)$

令 $%5Comega_k%3D%5Cfrac%7Bk%5Cpi%7D%7Bl%7D%2C%5C%3B%5CDelta%5Comega%3D%5Cfrac%7B%5Cpi%7D%7Bl%7D$ ，則有

$g(x)%3Da_0%2B%5Csum_%7Bk%3D1%7D%5E%5Cinfty(a_k%5Ccos%5Comega_k%20x%2Bb_k%5Csin%5Comega_k%20x)$

其中，

$%0A%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D%0Aa_%7Bk%7D%3D%5Cfrac%7B1%7D%7Bl%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(%5Cxi)%20%5Ccos%20%5Comega_k%20%5Cxi%20%5Cmathrm%7B~d%7D%20%5Cxi%20%5C%5C%0Ab_%7Bk%7D%3D%5Cfrac%7B1%7D%7Bl%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(%5Cxi)%20%5Csin%20%5Comega_k%20%5Cxi%20%5Cmathrm%7B~d%7D%20%5Cxi%0A%5Cend%7Barray%7D%5Cright.$

若 $l%5Crightarrow%5Cinfty$ ，則 $%5CDelta%5Comega%5Crightarrow%5Cmathrm%20d%20%5Comega%2C%5C%3B%20%5Comega_k$ 變?yōu)檫B續(xù)參量，以上各式取極限即

$%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D%0AA(%5Comega)%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%20%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(%5Cxi)%20%5Ccos%20%5Comega%20%5Cxi%20%5Cmathrm%7Bd%7D%20%5Cxi%20%5C%5C%0AB(%5Comega)%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%20%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(%5Cxi)%20%5Csin%20%5Comega%20%5Cxi%20%5Cmathrm%7Bd%7D%20%5Cxi%0A%5Cend%7Barray%7D%5Cright.(*)$

$f(x)%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20A(%5Comega)%20%5Ccos%20%5Comega%20x%20%5Cmathrm%7B~d%7D%20%5Comega%2B%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20B(%5Comega)%20%5Csin%20%5Comega%20x%20%5Cmathrm%7B~d%7D%20%5Comega$

上式稱為傅里葉積分，而（*）式稱為 $f(x)$ 的傅里葉變換式。

利用輔助角公式，上式還可以寫成：

$f(x)%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20C(%5Comega)%20%5Ccos%20%5B%5Comega%20x-%5Cphi(%5Comega)%5D%20%5Cmathrm%7B~d%7D%20%5Comega$

$C(%5Comega)$ 為振幅譜， $%5Cphi(%5Comega)$ 為相位譜。

以上只是形式結(jié)果，嚴(yán)謹(jǐn)?shù)臄?shù)學(xué)理論有：

當(dāng)分類討論 $f(x)$ 的奇偶性時(shí)，甚至可以反復(fù)橫跳：

傅里葉積分有復(fù)數(shù)形式。

$f(x)%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20F(%5Comega)%20e%5E%7Bi%20%5Comega%20x%7D%20d%20%5Comega$

$F(%5Comega)%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(x)%20e%5E%7B-%5Cmathrm%20i%20%5Comega%20x%7D%20%5Cmathrm%20d%20x$

可以寫為對(duì)稱形式：

$f(x)%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20F(%5Comega)%20e%5E%7Bi%20%5Comega%20x%7D%20d%20%5Comega$

$F(%5Comega)%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(x)%20e%5E%7B-%5Cmathrm%20i%20%5Comega%20x%7D%20%5Cmathrm%20d%20x$

并簡(jiǎn)記為：

$F(%5Comega)%3D%5Cmathscr%20F%5Bf(x)%5D%2C%5C%3Bf(x)%3D%5Cmathscr%20F%5E%7B-1%7D%5BF(%5Comega)%5D$

$f(x)$ 稱為原函數(shù)， $F(%5Comega)$ 稱為像函數(shù)。

傅里葉變換具有如下基本性質(zhì)：

$%5Cmathscr%20F%5Bf'(x)%5D%3D%5Cmathrm%20i%5Comega%20F(%5Comega)$

$%5Cmathscr%20F%5Cleft%5B%5Cint%5E%7B(x)%7Df(%5Cxi)%5Cmathrm%20d%5Cxi%5Cright%5D%3D%5Cfrac%7BF(%5Comega)%7D%7B%5Cmathrm%20i%20%5Comega%7D$

$%5Cmathscr%20F%5Bf(ax)%5D%3D%5Cfrac%7B1%7D%7Ba%7DF(%5Cfrac%7B%5Comega%7D%7Ba%7D)$

$%5Cmathscr%20F%5Bf(x-x_0)%5D%3DF(%5Comega)e%5E%7B-%5Cmathrm%20i%5Comega%20x_0%7D$

$%5Cmathscr%20F%5Bf(x)%20e%5E%7B%5Cmathrm%20i%5Comega_0%20x%7D%5D%3DF(%5Comega-%5Comega_0)$

$%5Cmathscr%20F%5Bf_1(x)*f_2(x)%5D%3D2%5Cpi%20F_1(%5Comega)F_2(%5Comega)$

其中 $f_%7B1%7D(x)%20*%20f_%7B2%7D(x)%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f_%7B1%7D(%5Cxi)%20f_%7B2%7D(x-%5Cxi)%20%5Cmathrm%7Bd%7D%20%5Cxi$ 稱為函數(shù) $f_1%2Cf_2$ 的卷積。

關(guān)于這一系列的定理，可能主要是卷積不太好懂...后面找個(gè)機(jī)會(huì)專門研究一下卷積...

多重傅里葉積分：

對(duì)于n維情況 $f(x_1%2Cx_2%2C...%2Cx_n)$ ，引入矢量 $%5Cvec%20k%20%3D%20(k_1%2Ck_2%2C...%2Ck_n)$ 則可以有

$f(%5Cvec%20r)%3D%5Ciiint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20F(%5Cvec%20k)e%5E%7B%5Cmathrm%20i%5Cvec%20k%20%5Ccdot%20%5Cvec%20r%7D%20%5Cmathrm%20d%20%5Cvec%20r$

$F(%5Cvec%20k)%3D%5Cfrac%7B1%7D%7B(2%5Cpi)%5E3%7D%5Ciiint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(%5Cvec%20r)e%5E%7B-%5Cmathrm%20i%5Cvec%20k%20%5Ccdot%20%5Cvec%20r%7D%20%5Cmathrm%20d%20%5Cvec%20k$

這里因?yàn)槲⒎?、積分運(yùn)算都是線性的，就可以簡(jiǎn)單推廣。矢量 $%5Cvec%20k$ （好像）就是我們平時(shí)見(jiàn)到的波矢。

5.3? $%5Cdelta$ ?函數(shù)

$%5Cdelta$ 函數(shù)是一種廣義函數(shù)，用于描述質(zhì)點(diǎn)、點(diǎn)電荷、瞬時(shí)沖量等理想模型，其定義如下：

$%5Cdelta(x)%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bc%7D%0A0%2C%5C%3Bx%5Cneq0%5C%5C%0A%5Cinfty%2C%5C%3Bx%3D0%0A%5Cend%7Barray%7D%5Cright.$ ,

$%5Cint_a%5Eb%5Cdelta(x)%5Cmathrm%20dx%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D%0A0%2C%5C%3Bab%3E0%5C%5C%0A1%2C%5C%3Ba%3C0%3Cb%0A%5Cend%7Barray%7D%5Cright.$

$%5Cdelta$ 函數(shù)是偶函數(shù)，其原函數(shù)是階躍函數(shù)：

$H(x)%3D%5Cint_%7B-%5Cinfty%7D%5Ex%5Cdelta(x)%5Cmathrm%20dx%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bc%7D%0A0%2C%5C%3Bx%3C0%5C%5C%0A1%2C%5C%3Bx%3E0%0A%5Cend%7Barray%7D%5Cright.$

它還有被稱為“挑選性”的性質(zhì)：對(duì)任意定義在 $%5Cmathbb%20R$ 的連續(xù)函數(shù) $f(%5Ctau)$ ,

$%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20f(%5Ctau)%20%5Cdelta%5Cleft(%5Ctau-t_%7B0%7D%5Cright)%20%5Cmathrm%7Bd%7D%20%5Ctau%3Df%5Cleft(t_%7B0%7D%5Cright)$