Fourier分析的那些事

2022-03-11 22:45 作者:子瞻Louis 0人讀過(guò) | 我要投稿

在我第一期專欄里，推導(dǎo)了一個(gè)周期為T(mén)的函數(shù)g在滿足一定條件時(shí)可以寫(xiě)為以下形式，

$g(u)%3D%5Ctilde%7Bg%7D%20(u)%3A%3D%5Cfrac%7Ba_0%7D2%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20a_n%5Ccos%20%5Cfrac%7B2%5Cpi%20nu%7D%7BT%7D%2Bb_n%5Csin%5Cfrac%7B2%5Cpi%20nu%7DT$

這里采用記號(hào) $%5Ctilde%20f$ 表示f的Fourier級(jí)數(shù)或積分以區(qū)分原本的函數(shù)f

不過(guò)有一個(gè)問(wèn)題，在函數(shù)的某些“特殊”的點(diǎn)處它是否收斂呢？若收斂它又收斂到多少呢？比如下面的函數(shù)

在x=1處它的Fourier級(jí)數(shù)是怎么樣的呢？實(shí)際上關(guān)于這個(gè)三角級(jí)數(shù)的逐點(diǎn)收斂性研究通常非常微妙，盡管它在近代函數(shù)論中占據(jù)了重要地位，但對(duì)有逐點(diǎn)收斂于它本身的三角級(jí)數(shù)表示的函數(shù)，這種函數(shù)類的內(nèi)部結(jié)構(gòu)至今也沒(méi)有描述清楚。不過(guò)本期當(dāng)然是不會(huì)討論太高深的問(wèn)題了

為了方便，，令 $x%3D%5Cfrac%7B2%5Cpi%20u%7DT$ ，于是得到 $%5Ctilde%20f(x)$ 是x的周期為2π的函數(shù)，

$%5Ctilde%20f(x)%3A%3D%5Ctilde%20g%5Cleft(%5Cfrac%20%7BTx%7D%7B2%5Cpi%7D%5Cright)%3D%5Cfrac%7Ba_0%7D2%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20a_n%5Ccos%20nx%2Bb_n%5Csin%20nx$

其中

$a_n%3D%5Cfrac1%5Cpi%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)%5Ccos%20nt%5Cmathrm%20dt%2C$

$b_n%3D%5Cfrac1%5Cpi%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)%5Csin%20nt%5Cmathrm%20dt$

Riemann-Lebesgue引理

下面將敘述的是一個(gè)十分重要的引理，它是研究Fourier級(jí)數(shù)逐點(diǎn)收斂性的基礎(chǔ)：

若局部可積函數(shù) $f$ 在區(qū)間 $(%5Calpha%2C%5Cbeta)$ 上（至少在反常積分意義下）絕對(duì)可積，則

$%5Clim_%7B%5Cmathbb%20R%5Cni%5Clambda%5Cto%5Cinfty%7D%5Cint_%5Calpha%5E%5Cbeta%20f(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%3D0$

這里 $%5Calpha%2C%5Cbeta$ 均可以取正無(wú)窮或負(fù)無(wú)窮

證? 因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=f" alt="f">在 $(%5Calpha%2C%5Cbeta)$ 上絕對(duì)可積，所以對(duì)任意 $%5Cepsilon%3E0$ ，可以找到一個(gè)區(qū)間 $%5Ba%2Cb%5D%5Csubset%20(%5Calpha%2C%5Cbeta)$ ，使得對(duì)任何 $%5Clambda%5Cin%5Cmathbb%20R$ 都有

$%5Cleft%7C%5Cint_%5Calpha%5E%5Cbeta%20f(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx-%5Cint_a%5Eb%20f(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%5Cright%7C%3C%5Cepsilon$

此時(shí) $f$ 在 $%5Ba%2Cb%5D$ 上是Riemann可積的，設(shè) $%5Ba%2Cb%5D$ 的一個(gè)分割

$%5C%7Ba%3Dx_0%3Cx_1%3C%5Cdots%3Cx_n%3Db%5C%7D$

記 $m_j%3A%3D%5Cinf_%7Bx%5Cin%5Bx_%7Bj-1%7D%2Cx_j%5D%7Df(x)$

引入 $%5Ba%2Cb%5D$ 上的分段常函數(shù)

$g(x)%3Dm_j%2Cx_%7Bj-1%7D%5Cle%20x%5Cle%20x_j$

由 $%5Ceta%20%3E0$ 在 $%5Ba%2Cb%5D$ 上Riemann可積，可得

$%5Cbegin%7Baligned%7D%5Cleft%7C%5Cint_a%5Eb%20f(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx-%5Cint_a%5Eb%20g(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%5Cright%7C%26%5Cle%5Cint_a%5Eb%20%7Cf(x)-g(x)%7C%7Ce%5E%7Bi%5Clambda%20x%7D%7C%5Cmathrm%20d%7C%5C%5C%26%5Cle%5Cint_a%5Eb%7Cf(x)-g(x)%7C%5Cmathrm%20dx%5C%5C%26%3C(b-a)%5Cepsilon_1%5Cend%7Baligned%7D$

又由于

$%5Cbegin%7Baligned%7D%5Cint_a%5Eb%20g(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%26%3D%5Csum_%7Bj%3D1%7D%5En%5Cint_%7Bx_%7Bj-1%7D%7D%5E%7Bx_j%7Dm_je%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%5C%5C%26%3D%5Cfrac1%7Bi%5Clambda%7D%5Csum_%7Bj%3D1%7D%5Enm_je%5E%7Bi%5Clambda%20x_j%7D-m_je%5E%7Bi%5Clambda%20x_%7Bj-1%7D%7D%5Cxrightarrow%7B%5Clambda%5Cto%5Cinfty%7D0%5Cend%7Baligned%7D$

于是便得到

$%5Cint_%5Calpha%5E%5Cbeta%20f(x)e%5E%7Bi%5Clambda%20x%7D%5Cmathrm%20dx%5Cxrightarrow%7B%5Clambda%5Cto%5Cinfty%7D0$

$%5Csquare%0A$

此外，分離實(shí)部與虛部還可得當(dāng) $%5Cmathbb%20R%5Cni%5Clambda%5Cto%5Cinfty$ 時(shí)

$%5Cint_%5Calpha%5E%5Cbeta%20f(x)%5Ccos%7B%5Clambda%20x%7D%5Cmathrm%20dx%5Cto0$

$%5Cint_%5Calpha%5E%5Cbeta%20f(x)%5Csin%7B%5Clambda%20x%7D%5Cmathrm%20dx%5Cto0$

Dirichlet核與局部化原理

根據(jù)Euler公式，F(xiàn)ourier級(jí)數(shù)可寫(xiě)為如下形式

$%5Ctilde%20f(x)%3D%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty%5Cleft(%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)e%5E%7B-int%7D%5Cmathrm%20dt%5Cright)e%5E%7Binx%7D$

該級(jí)數(shù)理解為柯西主值意義下的級(jí)數(shù)，即
$%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty%3A%3D%5Clim_%7BM%5Cto%5Cinfty%7D%5Csum_%7Bn%3D-M%7D%5EM$

取其部分和，

$%5Cbegin%7Baligned%7D%5Ctilde%20f_N(x)%26%3D%5Csum_%7Bn%3D-N%7D%5EN%5Cleft(%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)e%5E%7B-int%7D%5Cmathrm%20dt%5Cright)e%5E%7Binx%7D%5C%5C%26%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)%5Ccolor%7Bblue%7D%7B%5Csum_%7Bn%3D-N%7D%5ENe%5E%7Bin(x-t)%7D%7D%5Cmathrm%20dt%5Cend%7Baligned%7D$

記藍(lán)色部分為 $%5Cmathcal%20D_N(x-t)$ ，即

$%5Cmathcal%20D_N(t)%3D%5Csum_%7Bn%3D-N%7D%5ENe%5E%7Bint%7D$

稱它為Dirichlet核，顯然它有以下性質(zhì)：

$%5Cforall%20N%5Cin%5Cmathbb%20N%2CA%5Cin%5Cmathbb%20R%2C%5Cfrac1%7B2%5Cpi%7D%5Cint_A%5E%7BA%2B2%5Cpi%7D%5Cmathcal%20D_N(t)%5Cmathrm%20dt%3D1$
$%5Cmathcal%20D_N(t)%3D%5Cmathcal%20D_N(-t)$

利用這兩條性質(zhì)以及 $%5Ctilde%20f_N$ 的周期性，可得

$%5Ctilde%20f_N(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_0%5E%5Cpi(f(x%2Bt)%2Bf(x-t))%5Cmathcal%20D_N(t)%5Cmathrm%20dt$

對(duì)Dirichlet核繼續(xù)計(jì)算，可得

$%5Cbegin%7Baligned%7D%5Cmathcal%20D_N(t)%3D%5Csum_%7Bn%3D-N%7D%5ENe%5E%7Bint%7D%26%3D%5Cfrac%7Be%5E%7Bi(N%2B1)t%7D-e%5E%7BiNt%7D%7D%7Be%5E%7Bit%7D-1%7D%5C%5C%26%3D%5Cfrac%7Be%5E%7Bi(N%2B1%2F2)t%7D-e%5E%7B-i(N%2B1%2F2)t%7D%7D%7Be%5E%7Bit%2F2%7D-e%5E%7B-it%2F2%7D%7D%5C%5C%26%3D%5Cfrac%7B%5Csin%5Cleft(N%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac%2012t%7D%5Cend%7Baligned%7D$

代回至部分和中，可得

$%5Ctilde%20f_N(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_0%5E%5Cpi(f(x%2Bt)%2Bf(x-t))%5Cfrac%7B%5Csin%5Cleft(N%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac%2012t%7D%5Cmathrm%20dt$

然后可以試試將積分區(qū)間拆開(kāi)為 $%5B0%2C%5Cdelta)%5Ccup%5B%5Cdelta%2C%5Cpi%5D$ ，其中 $%5Cdelta%3E0$ ，

因?yàn)楫?dāng) $%5Cdelta%5Cle%20t%5Cle%5Cpi$ 時(shí)，有 $0%3C%5Csin%5Cfrac%5Cdelta2%5Cle%5Csin%5Cfrac%20t2$ ，所以根據(jù)Riemann-Lebesgue引理有

$%5Cint_%5Cdelta%5E%5Cpi(f(x%2Bt)%2Bf(x-t))%5Cfrac%7B%5Csin%5Cleft(N%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac%2012t%7D%5Cmathrm%20dt%5Cxrightarrow%7BN%5Cto%5Cinfty%7D0$

由此可得當(dāng) $N%5Cto%5Cinfty$ 時(shí)，

$%5Ctilde%20f_N(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_0%5E%5Cdelta(f(x%2Bt)%2Bf(x-t))%5Cfrac%7B%5Csin%5Cleft(N%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac%2012t%7D%5Cmathrm%20dt%2Bo(1)$

這個(gè)等式表明了函數(shù)的Fourier級(jí)數(shù)在一個(gè)點(diǎn)的收斂性完全取決于在以這個(gè)點(diǎn)為中心delta為半徑的鄰域內(nèi)的性質(zhì)，這就是所謂的局部化原理。由于 $%5Cdelta$ 可以任意小，所以局部化原理也可以簡(jiǎn)單表述為函數(shù)在x的任意小鄰域內(nèi)的性質(zhì)決定其Fourier級(jí)數(shù)的收斂性

觀察被積函數(shù)，可以較自然的引出一個(gè)定義：

$%5Cbar%20f(x)%3A%3D%5Clim_%7Bh%5Cto%2B0%7D%5Cfrac%7Bf(x%2Bh)%2Bf(x-h)%7D2$

接著就是結(jié)論了

Fourier級(jí)數(shù)收斂性

先給出兩個(gè)定義：

定義 $f$ 在x的左右極限：
$f(x%2B)%3A%3D%5Clim_%7Bh%5Cto%2B0%7Df(x%2Bh)%2C%5Cquad%20f(x-)%3A%3D%5Clim_%7Bh%5Cto%2B0%7Df(x-h)$
函數(shù) $f$ 在點(diǎn)x連續(xù)或第一類間斷，若對(duì)充分小的 $%5Cdelta%3E0$ ，存在 $M%3E0%2C0%3Ca%5Cle1$ ，使
$%7Cf(x%5Cpm%20t)-f(x%5Cpm)%7C%3CMt%5Ea%2C(0%3Ct%3C%5Cdelta)$
則稱 $f$ 在點(diǎn)x滿足H?lder條件，特別的，當(dāng)a=1是稱為Lipschitz條件

（定理）： $f%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$ 是周期為2π，在閉區(qū)間 $%5B-%5Cpi%2C%5Cpi%5D$ 上絕對(duì)可積的函數(shù)，若 $f$ 在點(diǎn)x滿足H?lder條件，則其Fourier級(jí)數(shù)在x收斂，且

$%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty%5Cleft(%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(t)e%5E%7B-int%7D%5Cmathrm%20dt%5Cright)e%5E%7Binx%7D%3D%5Cbar%20f(x)$

證?? $f$ 在點(diǎn)x滿足H?lder條件時(shí)，對(duì) $0%3Ct%3C%5Cdelta$

$%5Cfrac%7B%7Cf(x%5Cpm%20t)-f(x%5Cpm)%7C%7Dt%3CMt%5E%7B1-a%7D$

由此可得

$%5Cvarphi(t)%3A%3D%5Cfrac%7Bf(x%2Bt)-f(x%2B)%2Bf(x-t)-f(x-)%7Dt$

在區(qū)間 $%5B0%2C%5Cdelta%5D$ 上絕對(duì)可積。又有

$%5Ctilde%20f_N(x)-%5Cbar%20f(x)%3D%5Cfrac1%7B%5Cpi%7D%5Cint_0%5E%5Cdelta%5Cvarphi(t)%5Cfrac%7Bt%7D%7B2%5Csin%5Cfrac%2012t%7D%5Csin%5Cleft(N%2B%5Cfrac12%5Cright)t%5Cmathrm%20dt%2Bo(1)$

令 $%5Cdelta%5Cto0$ ，此時(shí) $2%5Csin%5Cfrac12t%5Csim%20t$ ，再由Riemann-Lebesgue引理可知上式當(dāng) $N%5Cto%5Cinfty$ 趨于零，即

$%5Clim_%7BN%5Cto%5Cinfty%7D%5Ctilde%20f_N(x)%3D%5Cbar%20f(x)$

$%5Csquare%0A$

該定理可以同樣簡(jiǎn)訴為滿足H?lder條件的函數(shù)的Fourier級(jí)數(shù)收斂于它任意小的鄰域中的左右平均值，不難發(fā)現(xiàn)它正好與局部化引理相對(duì)應(yīng)

還有最后一步，Dini-Lipschitz判別法只給出了周期為2π的函數(shù)其Fourier級(jí)數(shù)收斂的充分條件，但因?yàn)樽铋_(kāi)始我們用了代換 $x%3D%5Cfrac%7B2%5Cpi%20u%7DT$ ，這可以將任意一個(gè)周期為T(mén)的函數(shù)轉(zhuǎn)化為周期為2π的函數(shù)，所以我們實(shí)際上是得到了對(duì)任意周期函數(shù)其Fourier級(jí)數(shù)收斂的充分條件?，F(xiàn)在再將 $x%3D%5Cfrac%7B2%5Cpi%20u%7DT$ 代入回原式可得：

$%5Cbar%20f%5Cleft(%5Cfrac%7B2%5Cpi%20u%7D%7BT%7D%5Cright)%3D%5Cbar%7Bg%7D%20(u)%3D%5Cfrac%7Ba_0%7D2%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20a_n%5Ccos%20%5Cfrac%7B2%5Cpi%20nu%7D%7BT%7D%2Bb_n%5Csin%5Cfrac%7B2%5Cpi%20nu%7DT$

Fourier積分的收斂性

利用類似的方法，可將Fourier級(jí)數(shù)收斂的判別法推廣至Fourier積分，已知對(duì)非周期函數(shù)，當(dāng)它滿足一定條件時(shí)，有

$f(x)%3D%5Ctilde%20f(x)%3A%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20%5Cunderbrace%7B%5Cleft(%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-2%5Cpi%20i%5Comega%20t%7D%5Cmathrm%20dt%5Cright)%7D_%7B%5Chat%20f(%5Comega)%7De%5E%7B2%5Cpi%20i%5Comega%20x%7D%5Cmathrm%20d%5Comega$

這里的積分同樣也為主值意義下的積分，即

$%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%3A%3D%5Clim_%7BA%5Cto%5Cinfty%7D%5Cint_%7B-A%7D%5EA$

（定理） $f%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$ 是絕對(duì)可積的函數(shù)，若它在點(diǎn)x處滿足Lipschitz條件，則其Fourier積分在點(diǎn)x處收斂于f左右極限的平均

證? 取 $A%3E0$ ，

$%5Ctilde%20f_A(x)%3D%5Cint_%7B-A%7D%5EA%20%7B%5Cleft(%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-2%5Cpi%20i%5Comega%20t%7D%5Cmathrm%20dt%5Cright)%7De%5E%7B2%5Cpi%20i%5Comega%20x%7D%5Cmathrm%20d%5Comega$

因?yàn)閒絕對(duì)可積，所以這里可以合理的交換上式的積分次序，

$%5Cbegin%7Baligned%7D%5Ctilde%20f_A(x)%26%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)%5Cleft(%5Cint_%7B-A%7D%5EAe%5E%7B2%5Cpi%20i%5Comega(x-t)%7D%5Cmathrm%20d%5Comega%5Cright)%5Cmathrm%20dt%5C%5C%26%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)%5Cfrac%7B%5Csin2%5Cpi%20A(x-t)%7D%7B%5Cpi%20(x-t)%7D%5Cmathrm%20dt%5C%5C%26%3D%5Cint_0%5E%5Cinfty%20(f(x%2Bt)%2Bf(x-t))%5Cfrac%7B%5Csin%7B2%5Cpi%20At%7D%7D%7B%5Cpi%20t%7D%5Cmathrm%20dt%5Cend%7Baligned%7D$

因?yàn)镈irichlet積分

$%5Cint_%7B0%7D%5E%5Cinfty%5Cfrac%7B%5Csin%7B2%5Cpi%20At%7D%7D%7Bt%7D%5Cmathrm%20dt%3D%5Cfrac%5Cpi%202$

所以

$%5Cbar%20f(x)%3D%5Cint_%7B0%7D%5E%5Cinfty(f(x%2B)%2Bf(x-))%5Cfrac%7B%5Csin%7B2%5Cpi%20At%7D%7D%7B%5Cpi%20t%7D%5Cmathrm%20dt$

因此有

$%5Cbegin%7Baligned%7D%5Ctilde%20f_A(x)-%5Cbar%20f(x)%26%3D%5Cfrac1%5Cpi%5Cint_0%5E%5Cinfty%20%5Cvarphi(t)%7B%5Csin%7B2%5Cpi%20At%7D%7D%5Cmathrm%20dt%5C%5C%26%3D%5Cfrac1%5Cpi%5Cint_0%5E%5Cdelta%5Cvarphi(t)%5Csin2%5Cpi%20At%5Cmathrm%20dt%2B%5Cfrac1%5Cpi%5Cint_%5Cdelta%5E%5Cinfty%5Cvarphi(t)%5Csin2%5Cpi%20At%5Cmathrm%20dt%5Cend%7Baligned%7D$

其中

$%5Cvarphi(t)%3A%3D%5Cfrac%7Bf(x%2Bt)-f(x%2B)%2Bf(x-t)-f(x-)%7Dt$

f在點(diǎn)x滿足H?lder條件使得上式在 $%5B0%2C%5Cdelta%5D$ 上絕對(duì)可積，根據(jù)Riemann-Lebesgue引理可知

$%5Cfrac1%5Cpi%5Cint_0%5E%5Cdelta%5Cvarphi(t)%5Csin2%5Cpi%20At%5Cmathrm%20dt%5Cxrightarrow%7BA%5Cto%5Cinfty%7D0$

而第二項(xiàng)又可以寫(xiě)成 $f(x-t)%2Cf(x%2Bt)%2Cf(x%2B)%2Cf(x-)$ 的四個(gè)積分之和，對(duì)前兩個(gè)可以在此用Riemann-Lebesgue引理說(shuō)明他們趨于零，而后面兩個(gè)相對(duì)于積分是常因子，可以提到積分外，而根據(jù)Dirichlet積分的收斂性又可得后面兩個(gè)積分也趨于零，于是

$%5Clim_%7BA%5Cto%20%5Cinfty%7D%5Ctilde%20f_A(x)%3D%5Cbar%20f(x)%0A$

$%5Csquare%0A$

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Fourier分析的那些事

Riemann-Lebesgue引理

Dirichlet核與局部化原理

Fourier級(jí)數(shù)收斂性

Fourier積分的收斂性