In this paper we study the degree of approximation by superpositions of a sigmoidal function.We mainly consider the univariate case.If f is a continuous function,we prove that for any bounded sigmoidal function σ,d_...In this paper we study the degree of approximation by superpositions of a sigmoidal function.We mainly consider the univariate case.If f is a continuous function,we prove that for any bounded sigmoidal function σ,d_(n,σ)(f)≤‖σ‖ω(f,1/(n+1)).For the Heaviside function H(x),we prove that d_(n,H)(f)≤ω(f,1/(2(n+1))). If f is a continuous funnction of bounded variation,we prove that d_(n,σ)(f)≤‖σ‖/(n+1)V(f)and d_(n,H)(f)≤ 1/(2(n+1))V(f).For he Heaviside function,the coefficient 1 and the approximation orders are the best possible.We compare these results with the classical Jackson and Bernstein theorems,and make some conjec- tures for further study.展开更多
Polarization singularities,which emerge from the incoherent superposition of two vector electric fields with the same frequency,and their evolution in free space are studied analytically and illustrated by numerical e...Polarization singularities,which emerge from the incoherent superposition of two vector electric fields with the same frequency,and their evolution in free space are studied analytically and illustrated by numerical examples.It is shown that there exist C-points,L-lines,in particular,C-lines in incoherently superimposed two-dimensional wavefields.Usually,the C-lines are unstable and disappear during the free-space propagation.The motion,pair creation-annihilation process of the emergent C-points,as well as the distortion of the L-lines may take place,and the degree of polarization of the emergent C-points varies upon propagation and may be less than 1.展开更多
文摘In this paper we study the degree of approximation by superpositions of a sigmoidal function.We mainly consider the univariate case.If f is a continuous function,we prove that for any bounded sigmoidal function σ,d_(n,σ)(f)≤‖σ‖ω(f,1/(n+1)).For the Heaviside function H(x),we prove that d_(n,H)(f)≤ω(f,1/(2(n+1))). If f is a continuous funnction of bounded variation,we prove that d_(n,σ)(f)≤‖σ‖/(n+1)V(f)and d_(n,H)(f)≤ 1/(2(n+1))V(f).For he Heaviside function,the coefficient 1 and the approximation orders are the best possible.We compare these results with the classical Jackson and Bernstein theorems,and make some conjec- tures for further study.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10874125)
文摘Polarization singularities,which emerge from the incoherent superposition of two vector electric fields with the same frequency,and their evolution in free space are studied analytically and illustrated by numerical examples.It is shown that there exist C-points,L-lines,in particular,C-lines in incoherently superimposed two-dimensional wavefields.Usually,the C-lines are unstable and disappear during the free-space propagation.The motion,pair creation-annihilation process of the emergent C-points,as well as the distortion of the L-lines may take place,and the degree of polarization of the emergent C-points varies upon propagation and may be less than 1.
