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）︰2015/06/23 考試時限（分鐘）：180 試題 : 1. (a) (10%) Let f: |R → |R. Assume that f is Lebesgue measurable. Show that for any α ∈ |R, the set -1 f (α) is measurable. (1) (b) (20%) Is (1) sufficient to guarantee that f is a measurable function? If yes, prove the statement. If no, construct a counterexample, namely, find a non-measurable function f: |R → |R satisfying (1). n 2. (20%) We have shown in class that every measurable function f: |R → |R can be approximated by simple functions (pointwise a.e.). Show that any measurable function f can be approximated by step functions (also pointwise N a.e.). A step function s is defined as s = Σ α χ where I is a half- N N j=1 j I j n j open cube in |R and α ∈ |R. j 3. Let f be a periodic continuous function on [0, 2π] (period 2π). The Fourier series generated by f is a 0 ∞ f ~ ─ + Σ (a cos kx + b sin kx). 2 k=1 k k The n-th partial sum s is given by N a 0 N s = ─ + Σ (a cos kx + b sin kx). N 2 k=1 k k Show that 2 (a) (10%) s → f in L [0, 2π] (by Fejér's theorem and the best possible N approximation property). (b) (10%) The Fourier series can be integrated term by term, i.e., for all x we have a x x 0 ∞ x ∫ f(t)dt = ──+ Σ ∫ (a cos kt + b sin kt) dt, 0 2 k=1 0 k k the integrated series being uniformly convergence on every interval. (c) (10%) Show that 2 2 x π ∞ cos kx ─ = πx - ──+ 2 Σ ───, for 0 ≦ x ≦ 2π. 2 3 k=1 2 k 1 4. (a) (10%) If f ∈ C and is periodic of period 2π, then the Fourier series generated by f converges uniformly to f on [0, 2π]. (b) (10%) Use Plancherel's formula (or Parseval's identity) to derive 2 sin (αx) ∫ ─────dx = απ |R 2 x for any α > 0. Hint: consider the Fourier transform of χ . (-α, α) -- 2 2 1 ψxavier13540 給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則 ╰x(0) = x° t ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三 次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。 --

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