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複雑理工学専攻
1
単純な計算問題なので省略.
2
線形代数の基本的な問題です.
(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


Since the function



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


(18)
Hence we obtain
(19)
(3)
From the independence of

(20)
(4)
From the formula (18), we obtain
(21)