課程名稱︰分析導論優二
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/06/09
考試時限（分鐘）：35
試題 :
2 ∞ ikx ∞ ikx
1. (10%) Assume that f, g ∈ L [0, 2π] and f ~ Σ c e , g ~ Σ d e . Show
-∞ k -∞ k
that
1 2π ＿ ∞ ＿
──∫ fgdx = Σ c d .
2π 0 -∞ k k
2. (10%) Assume that f has continuous derivative on [0, 2π], f(0) = f(2π), and
2π
∫ f(t)dt = 0. Prove that
0
∥f∥ ≦ ∥f'∥,
where the equality holds iff f(x) = a cos x + b sin x. Here the norm is the
2
L [0, 2π] norm. (Use Parseval's formula.)