Integral over a subdivided interval

What do you mean by the '+' sign in the definition of phi_k. If it means that $f(x)_+ = \max(f(x), 0)$ (which is my best guess),

then for any particular i, there are not many values of j s.t. $\phi_i \phi_j \neq 0$ (and same for the derivatives).

That's correct.


Can you explain a bit further how you know this? I've just started reading about a new topic today so I may be unfamiliar with this kind of work.

Draw a sketch - each function is only non-zero over a small region. For the product to be non-zero the "non-zero" bits of each function will have to overlap.

Well, suppose i = j. What is $\phi_i^2$? There's an interval where it's $\left(1 - \frac{x - i / N}{1/ N}\right)^2$ (which you can make a little nicer by multiplying through by N), and another interval where it's $\left(1 + \frac{x - i / N}{1/ N}\right)^2$.

Neither of these is terribly difficult to integrate, and then you just add.

$\phi_i \phi_{i+1}$ is somewhat similar (but I think there's only one interval to consider - I haven't drawn the sketch).

You've not exactly posted much of your working. But almost certainly, you've got the limits of your integrals wrong.