We prove a generalized Gauss-Kuzmin-L′evy theorem for the generalized Gauss transformation Tp（x） = {p/x}.In addition, we give an estimate for the constant that appears in the theorem.
In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called du...In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂<sub>x</sub>) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.展开更多
We develop efficient and accurate sum-of-exponential(SOE)approximations for the Gaussian using rational approximation of the exponential function on the negative real axis.Six digit accuracy can be obtained with eigh...We develop efficient and accurate sum-of-exponential(SOE)approximations for the Gaussian using rational approximation of the exponential function on the negative real axis.Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms.This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform(FGT)in one and two dimensions.The one-dimensional scheme is particularly straightforward and easy to implement,requiring only twenty-four lines of MATLAB code.The two-dimensional version requires some care with data structures,but is significantly more efficient than existing FGTs.Following a detailed presentation of the theoretical foundations,we demonstrate the performance of the fast transforms with several numerical experiments.展开更多
文摘We prove a generalized Gauss-Kuzmin-L′evy theorem for the generalized Gauss transformation Tp（x） = {p/x}.In addition, we give an estimate for the constant that appears in the theorem.
文摘In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂<sub>x</sub>) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.
基金S.Jiang was supported in part by the United States National Science Foundation under grant DMS-1720405.
文摘We develop efficient and accurate sum-of-exponential(SOE)approximations for the Gaussian using rational approximation of the exponential function on the negative real axis.Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms.This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform(FGT)in one and two dimensions.The one-dimensional scheme is particularly straightforward and easy to implement,requiring only twenty-four lines of MATLAB code.The two-dimensional version requires some care with data structures,but is significantly more efficient than existing FGTs.Following a detailed presentation of the theoretical foundations,we demonstrate the performance of the fast transforms with several numerical experiments.
