Fourieranalysis : L^2 projection

Schwarzschild90 · Oct 12, 2016

Homework Statement

Textbook problem 14: find the L^2[-π, π] projection of f(x) = x^2 onto the space V_n spanned by 1/√(2π), sin(jx)/√π, cos(jx)/√π for j = 1 to n, then repeat for n = 2 and n = 3 and for g(x) = x^3.

2. and 3. Relevant equations and the attempt at a solution
We find the L^2 projection as such: [tex]<b_j , e_j >[/tex], where e_j is orthonormal basis j.

Now set [tex]b_j = < x^2 , e_j >[/tex] for [tex]1 \leq j \leq 3[/tex].

Ray Vickson · Oct 12, 2016

Schwarzschild90 said:

Homework Statement

View attachment 107327

2. and 3. Relevant equations and the attempt at a solution
We find the L^2 projection as such: [tex]<b_j , e_j >[/tex], where e_j is orthonormal basis j.

Now set [tex]b_j = < x^2 , e_j >[/tex] for [tex]1 \leq j \leq 3[/tex].

Is there a question in here somewhere?

Schwarzschild90 · Oct 12, 2016

Hi Ray. Yes and I've developed my solution much more. It's still not quite correct though.
Will someone help me?

Handwritten-style worked solution in Danish: orthogonal projection of f(x) = x^2 onto orthonormal basis V_n, using e1 = 1/sqrt(2pi), e2 = sin(2x)/sqrt(pi), e3 = cos(2x)/sqrt(pi) and integrals over -pi to pi giving the projection f0 = (1/3)sqrt(2pi^3)e1 + (1/sqrt(2pi))... with inner products computed.

Fourieranalysis : L^2 projection

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