Sobolev空间子集上的采样集条件

考虑借助采样不等式刻画Sobolev空间子集V_α中函数f的采样稳定性问题, 给出采样集X={x_j: j∈J}的刻画条件, 并证明在该条件下采样不等式成立, 即对任意的函数f∈V_α均可从其样本{f(x_j): j∈J}中被稳定地重构. 最后, 给出采样不等式上下界的显式表达式.

Sampling Set Conditions in Subsets of Sobolev Spaces

We considered the problem of the sampling stability of function f in the Sobolev space subset V_α by means of sampling inequality，and gave the characterization condition of the sampling set X={x_j: j∈J}, which proved that the sampling inequality held under this condition, in other words, for any function f∈V_α could be stably reconstructed from the sample {f(x_j): j∈J}. Finally, the explicit expressions of the upper and lower bounds of the sampling inequality were given.