We consider the three dimensional Cauchy problem for the Laplace equation{uxx（x,y,z）＋uyy（x,y,z）＋uzz（x,y,z）=0,x∈R,y∈R,0〈z≤,u（x,y,0）=g（x,y）x∈R,y∈R,uz（x,y,0）=0,x∈R,y∈R,where the data is given at z =...We consider the three dimensional Cauchy problem for the Laplace equation{uxx（x,y,z）＋uyy（x,y,z）＋uzz（x,y,z）=0,x∈R,y∈R,0〈z≤,u（x,y,0）=g（x,y）x∈R,y∈R,uz（x,y,0）=0,x∈R,y∈R,where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 〈 z 〈 1. The problem is ill-posed, the solution （if it exists） doesn＇t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.展开更多
This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Holder condition by wavelet methods.A point-wise lower bound estimation of that model is first investigated...This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Holder condition by wavelet methods.A point-wise lower bound estimation of that model is first investigated;then we provide a linear wavelet estimate to obtain the optimal convergence rate,which means that the convergence rate coincides with the lower bound.The non-linear wavelet estimator is introduced for adaptivity,although it is nearly-optimal.However,the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions,we finally discuss a data driven version without any assumptions on the estimated functions.展开更多
基金Supported by Beijing Natural Science Foundation (No.1092003) Beijing Educational Committee Foundation (No.00600054R1002)
文摘We consider the three dimensional Cauchy problem for the Laplace equation{uxx（x,y,z）＋uyy（x,y,z）＋uzz（x,y,z）=0,x∈R,y∈R,0〈z≤,u（x,y,0）=g（x,y）x∈R,y∈R,uz（x,y,0）=0,x∈R,y∈R,where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 〈 z 〈 1. The problem is ill-posed, the solution （if it exists） doesn＇t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.
基金supported by the National Natural Science Foundation of China(No.11901019)the Science and Technology Program of Beijing Municipal Commission of Education(No.KM202010005025).
文摘This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Holder condition by wavelet methods.A point-wise lower bound estimation of that model is first investigated;then we provide a linear wavelet estimate to obtain the optimal convergence rate,which means that the convergence rate coincides with the lower bound.The non-linear wavelet estimator is introduced for adaptivity,although it is nearly-optimal.However,the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions,we finally discuss a data driven version without any assumptions on the estimated functions.
