I'm preparing for my calculus exam.And I have one examle in my textbook that put me in a deadlock.
Suppose $f(x)$- piecewise continuous function on $[a,b]\in > \mathbb{R}$, $\{\phi_k(x)\}$ is orthogonal system of functions in space of piecewise continuous functions. Suppose that $f_k$-Fourier coefficients of function $f(x)$. What is the minimum value of function $$F(C_1,C_2,...,C_n)=\int_a^b\left(f(x)-\sum_{k=1}^nC_k\phi_k\right)^2dx.$$ And for what values of the argument, it is achieved?
I know a theorem about the extremal property of the Fourier series.
If Fourier series $\sum_kx^ke_k$ of vector $x\in X$ converges to vector $x_l\in X$, then $\forall y=\sum_{k=1}^\infty\alpha_ke_k$ we have $$||x-x_l||\leq||x-y||,$$ with equality when $y=x_l$
So, as I understand it, the minimum is achieved when $C_k$ equal to Fourier coefficients of function $f(x)$, but I don't understand what is the minimum value of function $F$.
Well, there are various ways of doing it. First, I will assume that the $\phi_k$ are orthonormal, not just orthogonal. Then just evaluate the integral and minimize directly.
Let me use the symbol $\hat{f}$ to denote the Fourier coefficients of $f$.
\begin{eqnarray} F(C) &=& \|f-\sum_{k=1}^n C_k \phi_k \|^2 \\ &=& \|f\|^2-2\sum_{k=1}^n C_k \langle f, \phi_k \rangle + \sum_{k=1}^n C_k^2 \\ & = & \|f\|^2-2\sum_{k=1}^n C_k \hat{f}_k + \sum_{k=1}^n C_k^2 \end{eqnarray} Now note that $F(\hat{f}) = \|f\|^2-\sum_{k=1}^n \hat{f}_k^2$, so we can write $F(C) = F(\hat{f})+ \sum_{k=1}^n |C_k-\hat{f}_k|^2$ (or $F(C) = F(\hat{f})+ \|C-\hat{f}\|^2$, using the $2$-norm on $\mathbb{R}^n$), from which the minimum value and the minimizing coefficients can be easily computed.