# Wavelet expansions

I'm looking into wavelets to approximate a known square-integrable function $$f(x) = \sum_{j,k} a_{j,k} \times 2^{j/2}\psi(2^j x - k), \qquad a_{j,k}=2^{j/2}\int f(x) \psi(2^jx-k)dx$$ and I'm happy with Haar wavelets $$\psi$$ to begin with, mostly because I can calculate the coefficients $$a_{j,k}$$ in closed form (see below). My 2 questions are:

1. How do I truncate the double sum over $$j,k$$? I naively tried $$-N\leq j,k\leq N$$ for some values of $$N$$ but that doesn't work for my displaced ramp input function $$f(x) = (x - c)^+$$ when $$c>1$$.
2. For the Haar wavelets, integration by parts yields $$a_{j,k} = -2^{j/2}[F(2^{-j}k) - F(2^{-j}(k + 1/2)) + F(2^{-j}(k+1))]$$ where $$F$$ is antiderivative of $$f$$. I did not see this result in any of my online readings, so I am curious why this might not be a useful result?

Thanks! p.