I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints?
Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ by elements from subset $M = \{ x \in H: \int\limits_0^1 x(t) dt = 1 \}$
I know that best approximation for this function is projector $P_M (a) = \sum\limits_{k=1}^{n}(x_k, a) x_k$, where $(x_k, a) = \int\limits_{0}^{1} x_k(t) a(t)dt$ is scalar product. But I can't get form of functions $x_k(t)$
You want to solve the problem \begin{align*} \text{Minimize} \quad & \int_0^1 [x(t) - a(t)]^2 \, \mathrm{d}t \\ \text{w.r.t.} \quad & x \in L^2(0,1) \\ \text{such that}\quad & \int_0^1 x(t) \, \mathrm{d}t = 1 \end{align*} The Lagrangian $L : L^2(0,1) \times \mathbb R \to \mathbb R$ is given by $$L(x, \lambda) = \int_0^1 [x(t) - a(t)]^2 \, \mathrm{d}t + \lambda \, \left( \int_0^1 x(t) \, \mathrm{d}t - 1\right).$$ Now, you have to solve the following system (the left-hand sides are the derivatives of $L$ w.r.t. $x$ and $\lambda$) for $x$ and $\lambda$: \begin{align*} 2 \, (x(t) - a(t) ) + \lambda &= 0 \quad\text{f.a.a. } t \in (0,1)\\ \int_0^1 x(t) \, \mathrm dt &= 0\end{align*} The solution of this system should be straight-forward.