In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are...In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are given. As an applica-is constructed.展开更多
This article considers weighted approximation of multivariate function in reproducing kernel Hilbert space, and gives a relation between nth minimal errors for standard and linear information in the randomized setting...This article considers weighted approximation of multivariate function in reproducing kernel Hilbert space, and gives a relation between nth minimal errors for standard and linear information in the randomized setting. Using this relation we can estimate the nth minimal error for standard information by the nth minimal error for linear information, and study the tractability and strong tractability for these two classes of information.展开更多
Let H=C^r,α([0,1]^d)be Hoelder space and G=L2)[0,1]^d)with the inner product given by <g,h>G=∫[0,1]^dg(x)h(x)dx ↓Ag,h∈G.This paper considers the embedding operator S:H→G,S(f)=f,f∈H.We prove that en(S,∧^s...Let H=C^r,α([0,1]^d)be Hoelder space and G=L2)[0,1]^d)with the inner product given by <g,h>G=∫[0,1]^dg(x)h(x)dx ↓Ag,h∈G.This paper considers the embedding operator S:H→G,S(f)=f,f∈H.We prove that en(S,∧^std)≤mink=0,1,…(ek(S,∧^all)^2+C·k/n·n^2(r+α)/d)^1/2,where en(S,∧^std)and en(S,∧^all)denote the nth minimal error of standard and linear information respectively in the worst case,average case and randomized settings,and C is a constant.展开更多
基金This work is supported by NNSF(10271022)of China.
文摘In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are given. As an applica-is constructed.
基金This work is supported by the National Natural Science Foundation of China (Grant No.: 10271001)
文摘This article considers weighted approximation of multivariate function in reproducing kernel Hilbert space, and gives a relation between nth minimal errors for standard and linear information in the randomized setting. Using this relation we can estimate the nth minimal error for standard information by the nth minimal error for linear information, and study the tractability and strong tractability for these two classes of information.
基金This research is supported by the National Natural Science Foundation of China(Grant No. 10271001).
文摘Let H=C^r,α([0,1]^d)be Hoelder space and G=L2)[0,1]^d)with the inner product given by <g,h>G=∫[0,1]^dg(x)h(x)dx ↓Ag,h∈G.This paper considers the embedding operator S:H→G,S(f)=f,f∈H.We prove that en(S,∧^std)≤mink=0,1,…(ek(S,∧^all)^2+C·k/n·n^2(r+α)/d)^1/2,where en(S,∧^std)and en(S,∧^all)denote the nth minimal error of standard and linear information respectively in the worst case,average case and randomized settings,and C is a constant.
