Convergence order of wavelet thresholding estimator for differential operators on Besov spaces

The almost everywhere convergence of wavelet series is important in wavelet analysis [S. Kelly, M.A. Kon, L.A. Raphael, Local convergence for wavelet expansions, J. Funct. Anal. 126 (1994) 102–138]. Since the thresholding method plays fundamental roles in data compression, signal processing and other areas, the pointwise convergence of resulting wavelet series was investigated by T. Tao and B. Vidakovic in 1996 and 2000. Chen and Meng study the same problem for differential operators in Lp(R) setting [Di-Rong Chen, Hong-Tao Meng, Convergence of wavelet thresholding estimators of differential operators, Appl. Comput. Harmon. Anal. 25 (2008) 266–275]. This paper deals with the uniform and norm convergence of wavelet thresholding estimator of differential operators on Besov spaces and Sobolev spaces with integer exponents. In particular, the convergence order is focused. To do that, we first study the corresponding problems for non-standard forms of those operators.