论文标题
尖锐的$ l_p $ -Error估算样本运营商
Sharp $L_p$-error estimates for sampling operators
论文作者
论文摘要
我们研究空间中线性采样运算符的近似属性$ l_p $，价格为$ 1 \ le P <\ infty $。通过Steklov的平均，我们引入了一种新的平滑度度量，同时包含有关$ L_P $中函数平滑度的信息，以及有关函数在采样点的行为的离散信息。新的平滑度衡量标准使我们能够将近似理论的几种经典结果改进并扩展到线性采样算子的情况下。特别是，我们在$ l_p $中获得采样操作员的匹配直接和反近似不平等，找到针对特定功能类别的相应$ l_p $ -Errors的确切衰减顺序，并引入一种特殊的$ K $功能及其实现，适合于研究采样操作员的平滑度。
We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_p$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_p$, find the exact order of decay of the corresponding $L_p$-errors for particular classes of functions, and introduce a special $K$-functional and its realization suitable for studying smoothness properties of sampling operators.
