Exercise 17, § 6

Let $\mu$ be Lebesgue measure on $X = [0, 1]$. One defines the sequence $(f_n)$ of step functions on $X$ by the following conditions: $f_0$ is the constant 1; for every integer $n > 0$, let $m$ be the largest integer such that $2^m \leq n$, and write $n = 2^m + k$; $f_n$ is the function equal to $2^{m/2}$ on the interval $[\frac{2k}{2^{m+1}}, \frac{2k+1}{2^{m+1}}]$, to $-2^{m/2}$ on the interval $[\frac{2k+1}{2^{m+1}}, \frac{2k+2}{2^{m+1}}]$, and to 0 at the other points of $X$.

a) Show that the sequence $(f_n)$ is orthonormal (the Haar orthonormal system).

b) Let $V_n$ be the linear subspace of $\mathscr{L}_C^2$ (over $\mathbf{C}$) generated by the $f_k$ with indices $k \leq n$. Show that there exists a partition of $[0, 1[$ into $n+1$ semi-open intervals such that, in each of these intervals, every function belonging to $V_n$ is constant. Deduce from this that, conversely, for every function $g$ that is constant on each of these $n+1$ intervals, there exists a function in $V_n$ that is equal to $g$ on $[0, 1[$ (note that $V_n$ has dimension $n+1$).

c) Let $g$ be any function in $\mathscr{L}_C^2$; deduce from b) that if $h$ is the unique function in $V_n$ for which $N_2(g-h)$ is minimal, then, in every interval $[\alpha, \beta[$ where $h$ is constant, one has $h(x) = \frac{1}{\beta-\alpha} \int_\alpha^\beta g(t)\, dt$.

d) Show that for every complex-valued function $g$, defined and continuous on $X$, the series with general term $(g|f_n)f_n(x)$ is uniformly convergent on $[0, 1[$ and has sum $g(x)$ (make use of c)). Deduce from this that the sequence $(f_n)$ is total.