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 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.
