I am bit unsure if I'm attacking the following problem correctly:
Let $n$ be a positive integer, and let $f$ be a continuous function defined on $[0,1]$. Let $h_k (t) = \sqrt{n} \phi(nt - k)$, where $\phi(t)$ is the Haar scaling function (which is $1$ on the interval $[0,1)$ and zero elsewhere). Form the $L^2$ projection of $f$ onto the span of the $h_k$'s,
$$f_n = \langle f,h_0 \rangle h_0 + . . . + \langle f, h_{n-1} \rangle h_{n-1}$$
Show that $f_n$ converges uniformly to $f$ on $[0,1]$.
OK, so I attempted to set this up as follows:
$$\langle f, h_k \rangle = \int_{-\infty}^{\infty} f(t) \sqrt{n} \phi(nt - k) dt$$
Since $\phi (nt - k)$ has value one on the interval $[k/n, k/n + 1/n]$ and zero elsewhere, we get:
$$\langle f, h_k \rangle = \sqrt{n} \int_{k/n}^{(k+1)/n} f(t) dt$$
However, I am unsure how to proceed from here, since we don't know anything more about $f(t)$. Of course I could then write:
$$\langle f, h_k \rangle h_k = n \phi(nt - k) \int_{k/n}^{(k+1)/n} f(t) dt$$
So:
$$f_n = n \sum_{k=0}^{n-1} \left( \phi(nt - k) \int_{k/n}^{(k+1)/n} f(t) dt \right)$$
But would this be a correct way to solve this problem? Or am I really off here? I haven't continued to the uniform convergence part yet, as I want to make sure that this first step is done correctly. If anyone can help me here, I would be very grateful!