Simultaneous Wavelet Deconvolution in Periodic Setting

. The paper proposes a method of deconvolution in a periodic setting which combines two important ideas, the fast wavelet and Fourier transform‐based estimation procedure of Johnstone et al. [J. Roy. Statist. Soc. Ser. B66 (2004) 547] and the multichannel system technique proposed by Casey and Walnut [SIAM Rev. 36 (1994) 537]. An unknown function is estimated by a wavelet series where the empirical wavelet coefficients are filtered in an adapting non‐linear fashion. It is shown theoretically that the estimator achieves optimal convergence rate in a wide range of Besov spaces. The procedure allows to reduce the ill‐posedness of the problem especially in the case of non‐smooth blurring functions such as boxcar functions: it is proved that additions of extra channels improve convergence rate of the estimator. Theoretical study is supplemented by an extensive set of small‐sample simulation experiments demonstrating high‐quality performance of the proposed method.