作者wheata (仙人指路为马)
看板NTU-Exam
标题[试题] 99上 王金龙 微积分甲上 第十二次小考
时间Sun Jun 5 03:15:02 2011
课程名称︰微积分甲上
课程性质︰必修
课程教师︰王金龙
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2011/1/6
考试时限（分钟）：30
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
A. (a) Let f(x) = |x| for x
ε [-π,π] and f( x + 2 ) = f(x) for all x.
Find the Fourier series of f.
(1;33mε：属於)
∞ sin(2k-1)
(b) Evaluate Σ ───── .
k=1 (2k-1)^3
B. Consider a vector space V = {f : [-π,π]→R | f is continuous} over
1 π
field R. We define the inner product 〈f,g〉= ─ ∫ f(x)g(x) dx and
π -π
________
the norm || f || = √〈f,f〉 for all f,g
ε V.
(a) Show that {1/√2, cos x, sin x, cos 2x, sin 2x,...}
is an orthonormal basis of V .
(b) Let Vn = span{1/√2, cos x, sin x, cos 2x, sin 2x,..., cos nx, sin nx},
which is a subspace of V. For any given fεV,show that the projection
of f on Vn is the n-th Fourier polynomial.
C. (a) Let fn be a sequence in V. If fn converges to f uniformly,
show that lim ||fn - f|| = 0.
n→∞
(a0)^2 ∞
(b) Show that ─── + Σ [(ak)^2 + (bk)^2] = || f ||^2, where ak, bk's
2 k=1
are the Fourier coefficient of f.
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※ 编辑: wheata 来自: 114.42.217.82 (06/05 03:16)
1F：推 samviga :均分10分/30分 的那一次啊XDD 06/08 03:59