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)