# Minimizing an integral function

I want to pick the $\mathbf c$ that minimize

$$\int_0^1 (x-0.5)^2 \left(\sum_i^N c_i F_i(x)\right)^2 \; \text{d}x$$ subject to the constraint $$\int_0^1 \sum_i^N c_i F_i(x) \; \text{d}x = 0$$

where $\mathbf F$ are some functions of $x$.

I have read that this can be minimized via solving the set of linear equations

$$\sum_i c_i \int_0^1 (x-0.5)^2 F_i(x) F_j(x) \; \text{d}x = 0, \qquad j=1\ldots N$$

but I'm not seeing how. Can someone please explain?

• What is $0dr$? Can you include where you found that set of linear equation? – Tyberius Sep 15 '17 at 19:08
• @Tyberius Sorry, that was a typo that I have now corrected. – rhombidodecahedron Sep 16 '17 at 3:47