Fourieranalysis : L^2 projection

Schwarzschild90
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Homework Statement


Opgave 14.PNG


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

Now set b_j = < x^2 , e_j > for 1 \leq j \leq 3.
 
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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: <b_j , e_j >, where e_j is orthonormal basis j.

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

Is there a question in here somewhere?
 
Hi Ray. Yes and I've developed my solution much more. It's still not quite correct though.
Will someone help me?
Opgave 14 a.PNG
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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