# 院試問題　東大　複雑理工学専攻　平成31年　専門基礎科目

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## 2

(1)
This is obvious.
(1) (2)
For , we have
(2) Therefore , and we obtain
(3) (3)
Since it holds that
(4) an eigenvector for the eigenvalue is
(5) Hence we obtain
(6) where .
Then it follows that
(7) and we obtain the desired result.

(4)
Since we have
(8) we can easily check that
(9) Here we used the fact .

(5)
From the problem (3) and (4), we have
(10) Hence we obtain
(11) (6)
It is obvious that .
By considering the cofactor expansion of with respect to the first column, we have
(12) Hence from the problem (5), we obtain
(13) ## 3

フーリエ変換の問題です. (4)以外は単純な計算問題です. (4)は の逆フーリエ変換を直接計算できるかと思いましたが, 計算がうまくいかなかったため, におけるフーリエ変換のユニタリ性(特に単射性)を用いました. 少し大げさな解き方かもしれません.

(1)
We denote the Fourier transform of as .

(14) (2)

(15) (3)
From the problem (1) and (2). we have from Fubini’s theorem

(16) (4)
It is not difficult to check that the Fourier transform of is .
Since the function is square-integrable on , we have from the unitarity of the Fourier transform .

## 4

(2)の冒頭で示す離散の場合の部分積分公式を使うと計算が楽です.

(1)

(17) (2)
Note that for every function and random variable , we have

(18) Hence we obtain

(19) (3)
From the independence of , we have

(20) (4)
From the formula (18), we obtain

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