A note on the wavelet oracle

The extent to which wavelet function estimators achieve benchmark levels of performance is sometimes described in terms of our ability to interpret a mythical oracle, who has access to the "truth" about the target function. Since he is so wise, he is able to threshold in an optimal manner - that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.