Convergence of wavelet thresholding estimators of differential operators

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given signal. The shrinkage method is applied in different areas, including data compression, signal processing and statistics. The almost everywhere convergence of resulting wavelet series has been established in [T. Tao, On the almost everywhere convergence of wavelet summation methods, Appl. Comput. Harmon. Anal. 3 (1996) 384–387] and [T. Tao, B. Vidakovic, Almost everywhere behavior of general wavelet shrinkage operators, Appl. Comput. Harmon. Anal. 9 (2000) 72–82]. With a representation of f′ in terms of wavelet coefficients of f, we are interested in considering the influence of wavelet thresholding to f on its derivative f′. In this paper, for the representation of differential operators in nonstandard form, we establish the almost everywhere convergence of estimators as threshold tends to zero.