Can adaptive estimators for Fourier series be of interest to wavelets?

There is a firm belief in the literature on statistical applications of wav elets that adaptive procedures developed for Fourier series, labelled by th at literature as 'linear', are inadmissible because they are created for es timation of smooth functions and cannot attain optimal rates of mean integr ated squared error convergence whenever an underlying function is spatially inhomogeneous, for instance, when it contains spikes/jumps and smooth part s. I use the recent remarkable results by Hall, Kerkyacharian and Picard on block-thresholded wavelet estimation to present a counterexample to that b elief.