Let be the collection of m-times continuously differentiable probability densities fon R<sup>d</sup> such that 丨D<sup>a</sup>f（x<sub>1</sub>）-D<sup>a</sup>f（x<su...Let be the collection of m-times continuously differentiable probability densities fon R<sup>d</sup> such that 丨D<sup>a</sup>f（x<sub>1</sub>）-D<sup>a</sup>f（x<sub>2</sub>）丨≤M‖x<sub>1</sub>-x<sub>2</sub>‖<sup>β</sup> for x<sub>1</sub>,x<sub>2</sub>∈R<sup>d</sup>,[a]=m,where D<sup>a</sup>denotes the differential operator defined by D<sup>a</sup>=（[a]）/（x<sub>1</sub><sup>a</sup>…x<sub>d</sub><sup>a</sup><sub>d</sub>）.Under rather weak conditionson K（x）,the necessary and sufficient conditions for sup丨<sub>n</sub>（x）-f（x）丨=0((（logn/n）<sup>λ</sup>/（d+3λ）,λ=m+β,f∈ are that ∫x<sup>a</sup>K（xi）dx=0 for 0【[a]≤m.Finally the convergenco rate at apoint is given.展开更多
基金The project supported by National Natural Science Foundation of China.
文摘Let be the collection of m-times continuously differentiable probability densities fon R<sup>d</sup> such that 丨D<sup>a</sup>f（x<sub>1</sub>）-D<sup>a</sup>f（x<sub>2</sub>）丨≤M‖x<sub>1</sub>-x<sub>2</sub>‖<sup>β</sup> for x<sub>1</sub>,x<sub>2</sub>∈R<sup>d</sup>,[a]=m,where D<sup>a</sup>denotes the differential operator defined by D<sup>a</sup>=（[a]）/（x<sub>1</sub><sup>a</sup>…x<sub>d</sub><sup>a</sup><sub>d</sub>）.Under rather weak conditionson K（x）,the necessary and sufficient conditions for sup丨<sub>n</sub>（x）-f（x）丨=0((（logn/n）<sup>λ</sup>/（d+3λ）,λ=m+β,f∈ are that ∫x<sup>a</sup>K（xi）dx=0 for 0【[a]≤m.Finally the convergenco rate at apoint is given.
