Let $f(x) = x^2$ on $[-\pi, \pi]$. Choose the real number $A$ to minimize $\int_{-\pi}^{\pi} |f(x)-A|^2dx$
So I see an integral with an absolute value squared term, so something like Bessel's inequality is jumping out at me. The difficultly is though that A must be a real number, so simply applying Bessel's here doesn't seem to work. My other idea was perhaps to construct a Fourier Series for A, then that should converge to a real number, and then apply Bessel's. However, I am not sure if this logic is correct.
Any help is appreciated, thank you!
We want to minimize $$\int_{-\pi}^\pi (x^2-A)^2 dx = \int_{-\pi}^\pi x^4 dx-2A \int_{-\pi}^\pi x^2 dx + A^2 \int_{-\pi}^\pi dx$$
This is equivalent to minimizing $ A^2 \int_{-\pi}^\pi dx -2A \int_{-\pi}^\pi x^2 dx $ or $$g(A) = \frac{A^2}{2} - \frac{\pi^2 }{3}A.$$
$$g^\prime(A) = A-\frac{\pi^2}{3} = 0.$$
$$A=\frac{\pi^2}{3}.$$
It is obvious that this is at the minimum (and not the maximum). Nevertheless $g^{\prime\prime}(A) =1 >0.$