推 samviga :均分10分/30分 的那一次啊XDD 06/08 03:59
課程名稱︰微積分甲上
課程性質︰必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/1/6
考試時限（分鐘）：30
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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：屬於)
∞ 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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※ 編輯: wheata 來自: 114.42.217.82 (06/05 03:16)