Question
Prove the partial Haar sum $P_M(f_N)$, defined by $P_M(f_N)=\sum_{0\leq j < M} Q_j(f_N)+P_0(f_N)$, decays exponentially in $M: ||f_N-P_M(f_N)||_{L^2([0,1))} = 2^{-M/2}$, where $f_N=2^N\chi_{[0,2^{-N}]}$.
$P_j$ is the expectation operator and $Q_j$ is the difference operator defined accordingly. The interval we talked about here are all dyadic intervals. \begin{equation*} P_jf(x):=\frac{1}{|I_j|}\int_{I_j}f(t)dt, \end{equation*} \begin{equation*} Q_jf(x):=P_{j+1}f(x)-P_jf(x), \end{equation*}
My idea
Calculate the Haar coefficients first, but \begin{equation*} \begin{aligned} <f_N,h_I> &= \int_I 2^N\chi_{[0,2^{-N}]} \frac{1}{\sqrt{|I|}}(\chi_{I_r}(x)-\chi_{I_l}(x)) dx = \frac{2^{N+1}}{\sqrt{|I|}} \int_{[2^{-N-1},2^{-N}]}\chi_{I_r}(x) dx \end{aligned} \end{equation*} and I am not sure how to simplify this Haar coefficients. There's no certain conditions given on the dyadic interval. But then I tried
\begin{equation*} \begin{aligned} ||f_N-P_M(f_N)||_{L^2([0,1))} &= ||f_N-\sum_{0\leq j < M} Q_j(f_N)-P_0(f_N)||_{L^2([0,1))} \\ &= ||f_N - P_{M+1}f_N||_{L^2([0,1))} \\ &= ||f_N - 2^{-M-1}\int_{I_{M+1}}f(t)dt||_{L^2([0,1))} \end{aligned} \end{equation*} where I got stuck again since I am not sure how to deal with the integral with a dyadic interval.