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

Abstract

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