→ ntust661:不知道這樣算不算 show @@ 03/27 16:03
y※ 引述《foodyoytk (元)》之銘言：
: 如題: A periodic function f(x)=f(x+T) is approximated by the finite sum of
: k
: it's Fourier series f(x)=Pk(x)=Ao+Σ[Ancos(nWox)+Bnsin(nWox)]
: n=1
: Wo=2π/T and the total mean square error is defined as
: T/2 2
: Ek=1/T ∫ [f(x)-Pk(x)] dx. If the coefficients in Pk(x) are determined by the
: -T/2
: Euler Formula,prove that the approximation has the "least total mean square
: error"property.
: 麻煩各位了＝ ＝
假如
k
f(x) = P (x) = Ao + Σ [An cos(ω nx) + Bn sin(ω nx)]
k n=1 o o
T T
收斂於 [-── , ──]
2 2
1
根據定義 ── ║f(x) - P (x)║ < ε
b-a k
1 T/2 2 2
= ─── ∫ f (x) - 2f(x)P (x) + P (x) dx
T -T/2 k k
打字方便，我把三角級數轉成廣義的Fourier級數 weighting fun. = 1
∞
P (x) = Σ ck ψk(x)
k 0
↑
Euler's Formula
b
< P (x) , f(x) > = ∫ P (x) f(x) dx
k a k
∞
= Σ ck < ψk , f(x) > ← Fourier Coefficient~~
0
∞ 2 2
= Σ ck ║ψk ║
0
2
= ║P (x)║
k
1 2 2
= ─── (║f(x)║ - ║P (x)║ ) < ε
T k
所以有 least total mean square error Property~~
最小 平均 平方
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