Prime dream（4）——Fejér定理

2022-03-19 16:33 作者:子瞻Louis 0人讀過 | 我要投稿

本系列文集：《Prime dream》

其他文集：《Analysis》《雜文集》

數(shù)學(xué)分析中的卷積是以反常積分來定義的： $f%2Cg%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$

$f*g(x)%3A%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)g(x-t)%5Cmathrm%20dt$

通常假設(shè)上式積分對? $x%5Cin%5Cmathbb%20R$ ?都存在，稱其為函數(shù) f 與 g 的卷積，通過積分變量代換可知卷積滿足對稱性

delta型函數(shù)族

假設(shè)現(xiàn)在有一根質(zhì)量為1的一維細(xì)線被放置在區(qū)間? $%5B0%2C%5Calpha%5D(%5Calpha%3E0)$ ?上，其密度是均勻分布的，那么它在點(diǎn)t的密度可由以下函數(shù)表示：

$%5Crho_%5Calpha(t)%3A%3D%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Brcl%7D%20%5Cfrac1%5Calpha%20%26%200%5Cle%20t%5Cle%20%5Calpha%20%5C%5C%200%20%26%20%5Ctext%7Botherwise%7D%20%5Cend%7Barray%7D%5Cright.$

它會隨著?α 的減小非零的區(qū)間越來越小，直到趨于零，同時(shí)非零區(qū)間的值也越來越大，但它始終滿足：

$%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5Crho_%5Calpha(t)%5Cmathrm%20dt%3D1$

令? $%5Calpha%5Cto0$ ?，就得到了Dirac函數(shù)

$%5Cforall%20t%E2%89%A00%2C%20%5Cdelta(t)%3D0$
$%5Cint_%7B%5Cmathbb%20R%7D%5Cdelta(t)%5Cmathrm%20dt%3D%5Cint_%7B-%5Cepsilon%7D%5E%5Cepsilon%5Cdelta(t)%5Cmathrm%20dt%3D1$

其中 $%5Cepsilon$ 是任意小的正數(shù)，我們能得到以下性質(zhì)：

$f*%5Cdelta(x)%3D%5Clim_%7B%5Calpha%5Cto0%7Df*%5Crho_%5Calpha(x)%3D%5Clim_%7B%5Calpha%5Cto0%7D%5Cfrac1%5Calpha%5Cint_%7Bx-%5Calpha%7D%5Ex%20f(t)%5Cmathrm%20dt$

容易發(fā)現(xiàn)若 f 在點(diǎn)x連續(xù)，則? $f*%5Cdelta(x)%5Cto%20f(x)$ ??，這啟發(fā)我們引入以下定義：

#）對依賴于參變量? $%5Calpha%5Cin%20A$ ?的函數(shù)? $%5CDelta_%5Calpha%3A%5Cmathbb%20R%5Cto%5Cmathbb%20R$ ?構(gòu)成的函數(shù)族? $%5C%7B%5CDelta_%5Calpha%2C%5Calpha%5Cin%20A%5C%7D$ ?，如果

$%5Cforall%20%5Calpha%5Cin%20A%2C%5CDelta_%5Calpha(x)%5Cge0$
$%5Cforall%20%5Calpha%5Cin%20A%2C%5Cint_%5Cmathbb%20R%5CDelta_%5Calpha%20(t)%5Cmathrm%20dt%3D1$
$%5Cforall%20%5Crho%3E0%2C%5Clim_%7B%5Calpha%5Cto%20%5Comega%7D%5Cint_%7B-%5Crho%7D%5E%5Crho%5CDelta_%5Calpha(t)%5Cmathrm%20dt%3D1$

那么就稱該函數(shù)族在? $%5Calpha%5Cto%5Comega$ ?時(shí)是delta型函數(shù)族，注意到由第一二個(gè)條件可推出第三個(gè)條件顯然等價(jià)于

$%5Clim_%7B%5Calpha%5Cto%20%5Comega%7D%5Cint_%7B%7Ct%7C%5Cge%5Crho%7D%5CDelta_%5Calpha(t)%5Cmathrm%20dt%3D0$

Dirac函數(shù)是? $%5Crho_%5Calpha$ ?中? $%5Calpha%5Cto0$ ?的結(jié)果，所以函數(shù)族? $%5C%7B%5Crho_%5Calpha%2C%5Calpha%5Cin%5Cmathbb%20R%5E%2B%5C%7D$ ?構(gòu)成的當(dāng)然是delta型函數(shù)族，然后再給出一個(gè)定義：

#’）函數(shù)? $f%3AI%5Cto%5Cmathbb%20C$ ?若滿足：

$x%5Cin%20E%5Csubset%20I%2C%5Cforall%20%5Cepsilon%3E0%2C%5Cexists%5Crho%3E0%2C%5Ctext%7Bs.t.%7D%5Cforall%20%7Cx-x'%7C%3C%5Crho%5CRightarrow%7Cf(x)-f(x')%7C%3C%5Cepsilon$

則稱 f 在 $E%5Csubset%20I$ ?上一致連續(xù)

第一眼看到它可能會想到這不就是連續(xù)的定義嘛，但仔細(xì)一想還是有所不同的——其實(shí)當(dāng)中的 $%5Crho%3D%5Crho(%5Cepsilon)$ 是只依賴于 $%5Cepsilon$ 的正數(shù)，而連續(xù)的標(biāo)準(zhǔn)定義中的ρ是和點(diǎn)x也有關(guān)的，這也說明了一致連續(xù)的函數(shù)必定是連續(xù)的，緊接著可以證明以下關(guān)于delta型函數(shù)族卷積的收斂性定理

（定理）有界函數(shù)? $f%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$ 在? $E%5Csubset%5Cmathbb%20R$ ?上一致連續(xù)，若對? $%5Calpha%5Cto%5Comega$ ?時(shí)的delta型函數(shù)族 $%5C%7B%5CDelta_%5Calpha%2C%5Calpha%5Cin%20A%5C%7D$ ，卷積? $%5Cforall%20%5Calpha%5Cin%5Cmathbb%20A%2Cf*%5CDelta_%5Calpha(x)$ ?存在，則

$x%5Cin%20E%2C%5Calpha%5Cto%5Comega%5Cquad%20%5CRightarrow%20f*%5CDelta_%5Calpha(x)%5Crightrightarrows%20f(x)$

證? 設(shè)在? $%5Cmathbb%20R$ 上 $%7Cf(x)%7C%5Cle%20M$ ，取? $%5Cepsilon%3E0%2C%5Crho%3D%5Crho(%5Cepsilon)%3E0$ ，對? $x%5Cin%20E%2C%5Calpha%5Cto%5Comega$ ?

$%5Cbegin%7Baligned%7D%26%7Cf*%5CDelta_%7B%5Calpha%7D(x)-f(x)%7C%5C%5C%3D%20%26%5Cleft%7C%5Cint_%7B%5Cmathbb%20R%7Df(x-t)%5CDelta_%5Calpha(t)%5Cmathrm%20dt-f(x)%5Cright%7C%3D%5Cleft%7C%5Cint_%7B%5Cmathbb%20R%7D(f(x-t)-f(x))%5CDelta_%5Calpha(t)%5Cmathrm%20dt%5Cright%7C%5C%5C%5Cle%20%26%5Cint_%7B-%5Crho%7D%5E%5Crho%7Cf(x-t)-f(x)%7C%5CDelta_%5Calpha(t)%5Cmathrm%20dt%2B%5Cint_%7B%7Ct%7C%5Cge%5Crho%7D%7Cf(x-t)-f(x)%7C%5CDelta_%5Calpha(t)%5Cmathrm%20dt%5C%5C%3C%26%5Cepsilon%2B2M%5Cint_%7B%7Ct%7C%5Cge%5Crho%7D%5CDelta_%7B%5Calpha%7D(t)%5Cmathrm%20dt%5Cend%7Baligned%7D$

由Delta型函數(shù)族的定義可知最后一個(gè)積分實(shí)際上是趨于零的，于是? $%5Cforall%5Cepsilon%2C%20%5Cepsilon'%3E0%2C%5Calpha%5Cto%5Comega$

$%7Cf*%5CDelta_%5Calpha(x)-f(x)%7C%3C%5Cepsilon%2B%5Cepsilon'$

對所有? $x%5Cin%20E$ ?都成立，即? $f*%5CDelta_%5Calpha$ ?當(dāng)? $%5Calpha%5Cto%5Comega$ ?時(shí)在 E 上一致收斂到? $f$

$%5Csquare$

Fejér定理

周期2π的函數(shù) f 的Fourier級數(shù)的部分和：

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

其Cesàro平均為

$%5Csigma_N(x)%3A%3D%5Cfrac%7BS_0(x)%2B%5Cdots%2BS_N(x)%7D%7BN%2B1%7D$

根據(jù)上一章我們將部分和寫為積分：

$S_n(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x-t)%5Cfrac%7B%5Csin%5Cleft(n%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac12t%7D%5Cmathrm%20dt$

于是有

$%5Csigma_N(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x-t)%5Cmathcal%20F_N(t)%5Cmathrm%20dt$

其中

$%5Cmathcal%20F_N(t)%3A%3D%5Cfrac1%7BN%2B1%7D%5Csum_%7Bn%3D0%7D%5EN%5Cfrac%7B%5Csin%5Cleft(n%2B%5Cfrac12%5Cright)t%7D%7B%5Csin%5Cfrac12t%7D$

稱為Fourier級數(shù)的Fejer核，利用積化和差公式，有

$%5Csin%5Cleft(n%2B%5Cfrac12%5Cright)t%5Csin%5Cfrac12t%3D%5Cfrac%7B%5Ccos%20nt-%5Ccos%20(n%2B1)t%7D2$

由此可得

$%5Cbegin%7Baligned%7D%5Cmathcal%20F_N(t)%26%3D%5Cfrac1%7BN%2B1%7D%5Ccdot%5Cfrac1%7B%5Csin%5E2%5Cfrac12t%7D%5Csum_%7Bn%3D0%7D%5EN%5Csin%5Cleft(n%2B%5Cfrac12%5Cright)t%5Csin%5Cfrac12t%5C%5C%26%3D%5Cfrac1%7BN%2B1%7D%5Ccdot%5Cfrac1%7B%5Csin%5E2%5Cfrac12t%7D%5Csum_%7Bn%3D0%7D%5EN%5Cfrac%7B%5Ccos%20nt-%5Ccos(n%2B1)t%7D2%5C%5C%26%3D%5Cfrac1%7BN%2B1%7D%5Ccdot%5Cfrac%7B1-%5Ccos(N%2B1)t%7D%7B2%5Csin%5E2%5Cfrac12t%7D%3D%5Cfrac1%7BN%2B1%7D%5Ccdot%5Cfrac%7B%5Csin%5E2%5Cfrac%7BN%2B1%7D2t%7D%7B%5Csin%5E2%5Cfrac12t%7D%5Cend%7Baligned%7D$

可以根據(jù)Dirichlet核的性質(zhì)，得到一個(gè)比較著名的積分：

$%5Cint_%7B-%5Cpi%7D%5E%5Cpi%5Cfrac%7B%5Csin%5E2%5Cfrac%7BN%2B1%7D2t%7D%7B%5Csin%5E2%5Cfrac12t%7D%5Cmathrm%20dt%3D2(N%2B1)%5Cpi$

接著定義函數(shù)

$%5Cmathfrak%20A_N(t)%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Brcl%7D%5Cfrac%7B%5Cmathcal%20F_N(t)%7D%7B2%5Cpi%7D%2C%20%26%20%7Ct%7C%5Cleq%202%5Cpi%5C%5C0%2C%20%26%20%7Ct%7C%3E2%5Cpi%5Cend%7Barray%7D%5Cright.$

可以驗(yàn)證該函數(shù)對正整數(shù)N組成的函數(shù)族在? $N%5Cto%5Cinfty$ ?時(shí)是delta型函數(shù)族：

$%5Cforall%20N%5Cin%5Cmathbb%20N%5E%2B%2C%5Cmathfrak%20A_N(t)%5Cge0$
$%5Cforall%20N%5Cin%5Cmathbb%20N%5E%2B%2C%5Cint_%7B%5Cmathbb%20R%7D%5Cmathfrak%20A_N(t)%5Cmathrm%20dt%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%5Cmathcal%20F_N(t)%5Cmathrm%20dt%3D1$
$%5Cforall%20%5Crho%3E0%2C%5Cint_%7B%7Ct%7C%5Cge%5Crho%7D%5Cmathfrak%20A_N(t)%5Cmathrm%20dt%3D%5Cfrac1%7B%5Cpi%7D%5Cint_%7B%5Crho%7D%5E%5Cpi%5Cmathcal%20F_N(t)%5Cmathrm%20dt%5Cle%5Cfrac1%7B%5Cpi(N%2B1)%7D%5Cint_%7B%5Crho%7D%5E%5Cpi%5Cfrac%7B%5Cmathrm%20dt%7D%7B%5Csin%5E2%5Cfrac12t%7D%5Cxrightarrow%7BN%5Cto%5Cinfty%7D0$

又因?yàn)?/p>

$%5Csigma_N(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x-t)%5Cmathcal%20F_N(t)%5Cmathrm%20dt%3Df*%5Cmathfrak%20A_N(x)$

所以由delta型函數(shù)卷積的收斂性定理可以得到以下定理

（Fejér定理） $f%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$ 是? $%5B-%5Cpi%2C%5Cpi%5D$ 上絕對可積，周期為2π的函數(shù)，若?f?在? $E%5Csubset%5Cmathbb%20R$ 上一致連續(xù)，則

$x%5Cin%20E%2CN%5Cto%5Cinfty%2C%5Cquad%20%5CRightarrow%20%5Csigma_N(x)%5Crightrightarrows%20f(x)$

微積分中的Cauchy命題表明，F(xiàn)ourier級數(shù)部分和的極限若存在，則它與它的Cesàro平均收斂到相同的極限，因此函數(shù) f 的Fourier級數(shù)在它的連續(xù)點(diǎn)處要么發(fā)散，要么收斂到它本身

通過類似上一末結(jié)尾的操作，可以用變量代換將該結(jié)論推廣到任何周期函數(shù)

Fourier積分的Fejér定理

用類似的方法將Fourier級數(shù)推廣至Fourier積分，對滿足一定條件的函數(shù) f?，其Fourier積分為

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

作代換 $%5Cxi%3D2%5Cpi%5Comega$ ，可得

$I(x)%3D%5Cfrac1%7B2%5Cpi%20%7D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20%7B%5Cleft(%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-i%5Cxi%20t%7D%5Cmathrm%20dt%5Cright)%7De%5E%7Bi%5Cxi%20x%7D%5Cmathrm%20d%5Cxi$

為了方便，記

$F(%5Cxi)%3A%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-i%5Cxi%20t%7D%5Cmathrm%20dt$

取 I 的絕對值不超過A的積分

$I_A(x)%3A%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-A%7D%5EAF(%5Cxi)e%5E%7Bi%5Cxi%20x%7D%5Cmathrm%20d%5Cxi$

其積分平均為

$%5Cmathfrak%20S_T(x)%3D%5Cfrac1T%5Cint_0%5ETI_A(x)%5Cmathrm%20dA$

畫出這個(gè)二重積分的積分區(qū)域

由此交換求和順序可得

$%5Cbegin%7Baligned%7D%5Cmathfrak%20S_T(x)%26%3D%5Cfrac1T%5Cint_%7B-T%7D%5ET%5Cint_%7B%7C%5Cxi%7C%7D%5ETF(%5Cxi)e%5E%7Bi%5Cxi%20x%7D%5Cmathrm%20dA%5Cmathrm%20d%5Cxi%5C%5C%26%3D%5Cint_%7B-T%7D%5ET%5Cleft(1-%5Cfrac%7B%7C%5Cxi%7C%7DT%5Cright)F(%5Cxi)e%5E%7Bi%5Cxi%20x%7D%5Cmathrm%20d%5Cxi%5Cend%7Baligned%7D$

再由F的定義，有

$%5Cmathfrak%20S_T(x)%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)%5Cmathfrak%20F_T(x-t)%5Cmathrm%20dt$

其中? $%5Cmathfrak%20F_T$ ?是積分Fejer核，

$%5Cmathfrak%20F_T(u)%3A%3D%5Cint_%7B-T%7D%5ET%5Cleft(1-%5Cfrac%7B%7C%5Cxi%7C%7DT%5Cright)e%5E%7Bi%5Cxi%20u%7D%5Cmathrm%20d%5Cxi$

通過分部積分可以算得

$%5Cmathfrak%20F_T(u)%3DT%5Cleft(%5Cfrac%7B%5Csin%20Tu%2F2%7D%7BTu%2F2%7D%5Cright)%5E2$

令? $k_T%3A%3D%5Cfrac%7B%5Cmathfrak%20F_T%7D%7B2%5Cpi%7D$ ，則

$%5Cmathfrak%20S_T(x)%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)k_T(x-t)%5Cmathrm%20dt%3Df*k_T(x)$

首先顯然有? $k_T$ ?非負(fù)，再由其定義，我們可以引入以下函數(shù)：

$%5CPhi_T(%5Cxi)%3A%3D%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Brcl%7D%20%5Cleft(1-%5Cfrac%7B%7C%5Cxi%7C%7D%7BT%7D%5Cright)%2C%20%26%7C%5Cxi%7C%5Cle%20T%20%5C%5C0%2C%20%26%20%7C%5Cxi%7C%3ET%5Cend%7Barray%7D%20%5Cright.$

顯然它滿足Lipschitz條件，因此可以將其寫為Fourier積分，

$%5Cbegin%7Baligned%7D%5CPhi_T(w)%26%3D%5Cfrac1%7B2%5Cpi%7D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5Ccolor%7Bred%7D%7B%5Cleft(%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5CPhi_T(%5Cxi)e%5E%7Bi%5Cxi%20u%7D%5Cmathrm%20d%5Cxi%5Cright)%7De%5E%7B-iwu%7D%5Cmathrm%20du%5C%5C%26%3D%5Cint_%7B%5Cmathbb%20R%7Dk_T(u)e%5E%7Biwu%7D%5Cmathrm%20du%5Cend%7Baligned%7D$

取 w=0 ，可得對任意? $T%3E0$ ，

$%5Cint_%5Cmathbb%20R%20k_T(u)%5Cmathrm%20du%3D1$

與此同時(shí)又得到了一個(gè)有用的積分：

$%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5Cleft(%5Cfrac%7B%5Csin%20Tu%2F2%7D%7BTu%2F2%7D%5Cright)%5E2%5Cmathrm%20du%3D%5Cfrac%7B2%5Cpi%7D%7BT%7D$

又有對? $%5Crho%3E0$ ?，當(dāng)? $T%5Cto%5Cinfty$ 時(shí)

$%5Cint_%7B%7Cu%7C%5Cge%5Crho%7D%20k_T(u)%5Cmathrm%20du%3D%5Cfrac1%5Cpi%5Cint_%7B%5Crho%7D%5E%5Cinfty%5Cmathfrak%20F_T(u)%5Cmathrm%20du%5Cle%5Cfrac4%7BT%5Cpi%20%7D%5Cint_%7B%5Crho%7D%5E%5Cinfty%5Cfrac1%7Bu%5E2%7D%5Cmathrm%20du%5Cxrightarrow%7BT%5Cto%5Cinfty%7D0$

這說明了? $k_T$ ?組成的函數(shù)族在? $T%5Cto%5Cinfty$ ?是delta型函數(shù)族，于是由其卷積的收斂性定理，可得Fourier積分的Fejér定理：

（Fejér定理） $f%3A%5Cmathbb%20R%5Cto%5Cmathbb%20C$ ?是? $%5Cmathbb%20R$ 上絕對可積的函數(shù)，若? $f$ ?在? $E%5Csubset%5Cmathbb%20R$ ?上一致連續(xù)，則

$x%5Cin%20E%2CT%5Cto%5Cinfty%5Cquad%5CRightarrow%20%5Cmathfrak%20S_T(x)%5Crightrightarrows%20f(x)$

結(jié)語

這期我們由卷積與Dirac函數(shù)引入了delta型函數(shù)族，并證明了滿足某種條件時(shí)，它與函數(shù)的卷積收斂于該函數(shù)，于是得以證明了fourier分析中的Fejér定理，這個(gè)定理將會在下一期素?cái)?shù)定理（較弱形式）的證明中用到，沒錯(cuò)，正是數(shù)論中大名鼎鼎的素?cái)?shù)定理，盡管這個(gè)定理看上去與素?cái)?shù)毫無聯(lián)系，但它們就是存在如此微妙的聯(lián)系——這就是數(shù)學(xué)，不是么？

參考

《數(shù)學(xué)分析》 by B.A.卓里奇
《Fourier Analysis》 by Javier Duoandikoetxea (writ.), David Cruz
《數(shù)論導(dǎo)引》by?華羅庚

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delta型函數(shù)族

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結(jié)語