Almost Everywhere Behavior of General Wavelet Shrinkage Operators

AbstractWavelet shrinkage estimators are obtained by applying a shrinkage rule on the empirical wavelet coefficients. Such simple estimators are now well explored and widely used in wavelet-based nonparametrics. Results of Tao (1996, Appl. Comput. Harmon. Anal.3, 384–387) demonstrated that hard and soft thresholding shrinkage estimators absolutely converge almost everywhere to the original function when the threshold value goes to zero. Such natural and intuitive behavior of threshold estimators is expected, yet this result does not translate to the Fourier expansions. In this paper we show that almost everywhere convergence of shrinkage estimators holds for a range of shrinkage rules, not necessarily thresholding, subject to some mild technical conditions. Comments about the norm convergence are provided as well