In their classic paper "Ideal spatial adaptation by wavelet shrinkage" (http://biomet.oxfordjournals.org/content/81/3/425.short?rss=1&ssource=mfr), Donoho and Johnstone make the following statements:

Let $f$ be a piecewise polynomial function. If the list of true discontinuities in $f$ was known via an oracle, we could use a least squares estimate for each of the segments to obtain an estimate $\hat f$ with ideal risk $\sigma^2 L(D+1)$, where $L$ is the number of segments and $D$ the degree of the polynomial. They call this "piecewise polynomial adaptation".

Later on, they consider a hypothetical method called "selective wavelet reconstruction". For each wavelet function $W_{j,k}$, and the wavelet coefficients

$w_{j,k} = \theta_{j,k} + z_{j,k}$

where $\theta$ and $z$ denote the signal and noise part of the coefficient, this method obtains a subset $\delta^\star$ of wavelet coefficient indices $(j,k)$ from an oracle such that a wavelet is in the list iff $\theta_{j,k} \neq 0$. In other words, it keeps those wavelets that span a discontinuity in $f$, hence removing all coefficients that are due to noise only.

They then go on to claim that

because of the orthogonality of the $(W_{j,k})$, $\sum_{(j,k)\in \delta^\star} w_{j,k} W_{j,k}$ is the least-squares estimate of $f$ and [the risk] $R(T(y, \delta^\star), f) = n^{-1}\{\#(\delta^\star)\}\sigma^2$

I do understand how this risk is derived (the retained coefficients are still contaminated by noise), but I don't see why selective wavelet reconstruction would be an LSE? The risk they derive for this method is similar, but not equal to the one for polynomial adaptation, which clearly IS an LSE.

I am clearly missing something here. Could somebody please explain how one would derive the second method to yield an LSE, and why that would be different from polynomial adaptation via least squares?