This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavel...This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over Lprisks for p 2, and a nearly-optimal result for 1 < p < 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic(1999), Butucea and Tsybakov(2008), Comte et al.(2006), Lacour(2006) and Lounici and Nickl(2011).展开更多
Precision is one of the most important aspects of manufacturing.High precision creates high quality,high performance,exchangeability,reliability,and added value for industrial products.Over the past decades,remarkable...Precision is one of the most important aspects of manufacturing.High precision creates high quality,high performance,exchangeability,reliability,and added value for industrial products.Over the past decades,remarkable advances have been achieved in the area of high-precision manufacturing technologies,where the form accuracy approaches the nanometer level and surface roughness the atomic level.These extremely high precision manufacturing technologies enable the development of high-performance optical elements,semiconductor substrates,biomedical parts,and so on,thereby enhancing the ability of human beings to explore the macroand microscopic mysteries and potentialities of the natural world.In this paper,state-of-the-art high-precision material removal manufacturing technologies,especially ultraprecision cutting,grinding,deterministic form correction polishing,and supersmooth polishing,are reviewed and compared with insights into their principles,methodologies,and applications.The key issues in extreme precision manufacturing that should be considered for future R&D are discussed.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 11526150, 11601383 and 11271038)
文摘This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over Lprisks for p 2, and a nearly-optimal result for 1 < p < 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic(1999), Butucea and Tsybakov(2008), Comte et al.(2006), Lacour(2006) and Lounici and Nickl(2011).
基金funded by National Natural Science Foundation of China(50805039)Doctoral Fund of Ministry of Education of China(20070213049)+1 种基金Aero Science Foundation of China(2009ZE77008)Research Innovation Foundation in Harbin Institute of Technology(HIT.NSRIF.2008.40)~~
文摘Precision is one of the most important aspects of manufacturing.High precision creates high quality,high performance,exchangeability,reliability,and added value for industrial products.Over the past decades,remarkable advances have been achieved in the area of high-precision manufacturing technologies,where the form accuracy approaches the nanometer level and surface roughness the atomic level.These extremely high precision manufacturing technologies enable the development of high-performance optical elements,semiconductor substrates,biomedical parts,and so on,thereby enhancing the ability of human beings to explore the macroand microscopic mysteries and potentialities of the natural world.In this paper,state-of-the-art high-precision material removal manufacturing technologies,especially ultraprecision cutting,grinding,deterministic form correction polishing,and supersmooth polishing,are reviewed and compared with insights into their principles,methodologies,and applications.The key issues in extreme precision manufacturing that should be considered for future R&D are discussed.
