課程名稱︰實分析二
課程性質︰數學研究所必修課，應用數學研究所必選課
課程教師︰沈俊嚴
開課學院：理學院
開課系所︰數學研究所
考試日期︰2018年06月27日(三)
考試時限：10:20-12:10，110分鐘
試題 :
REAL ANALYSIS II FINAL EXAM. 2018/06/27
Do the following problems and write your arguments as detail as possible
1
1. (15%) Let m be the usual Lebesgue measure on |R, and define
1
μ(E) = ∫ ----- dm(x),
E 4
1+x
for Lebesgue measurable set E. Show that m is absolutely continuous with
respect to μ, and compute the Radon-Nikodym derivative dm/dμ.
1
2. (15%) Let l be a bounded linear functional on L[-1,1] such that l(f) = 0
1
as long as f is an odd function in L[-1,1]. Show that there exists an even
1 1
function g so that l(f) = ∫f(x)g(x)dx, for all f(x)∈L[-1,1].
-1
1
3. (15%) Let m be the usual Lebesgue measure on |R. Now define two measures
m and m by m(E) = m(E∩[0,1]) and m(E) = χ(0).
1 2 1 2 E
(1): Show that ∫fdm = f(0)m(E).
E 2 2
(2): Is m or m absolutely continuous or singular with respect to m?
1 2
1
4. (15%) Let μ be a finite Borel measure on |R, and define f(x) =μ((-∞,x]).
μ
Show that f(x) is continuous from the right.
μ
n
5. (15%) If A⊂|R, define the Hausdorff dimension of A as follows: If H(A)=0
α
for all α>0, let dim A = 0; otherwise, let dim A = sup{α:H(A)=∞}.
α
(1) Show that H(A) = 0 if α > dim A and H(A) = ∞ if α < dim A.
α α
(2) Show that every countable set has Hausdorff dimension 0.
6. (15%) Let g(x) be periodic and equal to log(1/|2sin(x/2)|) in (-π,π),
and let g(0) = 0. Compute explicitly its Fourier series
a0 ∞
---- + Σ ak cos(kx) + bk sin(kx).
2 k=1
In other words, compute explicitly the values of a0, ak, bk.
7. (10%) Let f(x) be periodic and integrable in (-π,π). Suppose f(x) is
continuous at x0, show that the arithmetic means σn(x) of its Fourier
series converge to f(x0) at x=x0. Hint: You may want to use the formula
1 π 2 sin[(n+1)t/2] 2
that σn(x) = ---- ∫ f(x+t)K(t)dt, where Kn(t) = ----- (-------------).
π -π n n+1 2sin(t/2)
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