Solve 2 orthogonal function on the interval?

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2014-05-11 7:10 am

The functions f1(x) = x and f2(x) = x^6 are orthogonal on the interval [−2, 2]. Find constants c1 and c2 such that f3(x) = x + c1x^2 + c2x^3 is orthogonal to both f1 and f2 on the same interval.

c1=?
c2=?

回答 (1)

2014-05-11 8:41 am

✔ 最佳答案

Here, the inner product is <f(x), g(x)> = ∫(x = -2 to 2) f(x) g(x) dx.

(i) Orthogonality to x:
We need ∫(x = -2 to 2) x * (x + c₁ x^2 + c₂ x^3) dx = 0
==> ∫(x = -2 to 2) (x^2 + c₁ x^3 + c₂ x^4) dx = 0
==> (x^3/3 + c₁ x^4/4 + c₂ x^5/5) {for x = -2 to 2} = 0
==> 16/3 + 0 + 64c₂/5 = 0
==> c₂ = -5/12.

(ii) Orthogonality to x^6:
We need ∫(x = -2 to 2) x^6 * (x + c₁ x^2 + c₂ x^3) dx = 0
==> ∫(x = -2 to 2) (x^7 + c₁ x^8 + c₂ x^9) dx = 0
==> (x^8/8 + c₁ x^9/9 + c₂ x^10/10) {for x = -2 to 2} = 0
==> c₁ * 1024/9 = 0
==> c₁ = 0.

I hope this helps!

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