[試題] 103下 王振男 分析導論優二 期末考

[xavier13540 (柊 四千)]

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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