Asymptotic minimaxity of wavelet estimators with sampled data

Donoho and Johnstone (1998) studied a setting where data were obtained in t he continuum white noise model and shop;ed that scalar nonlinearities appli ed to wavelet coefficients gave estimators which were asymptotically minima x over Besov balls. They claimed that this implied similar asymptotic minim axity results in the sampled-data model. In this paper we carefully develop and fully prove this implication. Our results are based on a careful definition of an empirical wavelet trans form and precise bounds on the discrepancy between empirical wavelet coeffi cients and the theoretical wavelet coefficients.