论文标题
具有较高混合规律性的功能的自适应抽样恢复
Adaptive sampling recovery of functions with higher mixed regularity
论文作者
论文摘要
我们将Faber样条系统从[14]张开，以证明具有较高混合规律性的多元功能空间的序列空间同构。相应的基础系数是与经典Faber Schauder系统相似的离散函数值的局部线性组合。这样可以通过仅存储离散（有限）函数值来使用截断的串联扩展来稀疏表示函数。获取函数值的节点集取决于以非线性方式的相应函数。确实，如果我们自适应地选择基本函数，则与双曲线交叉投影相比，要代表初始功能$ \ varepsilon> 0 $（例如$ l_ \ infty $）。此外，由于Faber花纹的规律性较高，我们克服了（混合）平滑度限制$ r <2 $，并受益于较高的混合规律性。作为副产品，我们为多元设置的Triebel专着[46]中的问题3.13的解决方案。
We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values similar as for the classical Faber Schauder system. This allows for a sparse representation of the function using a truncated series expansion by only storing discrete (finite) set of function values. The set of nodes where the function values are taken depends on the respective function in a non-linear way. Indeed, if we choose the basis functions adaptively it requires significantly less function values to represent the initial function up to accuracy $\varepsilon>0$ (say in $L_\infty$) compared to hyperbolic cross projections. In addition, due to the higher regularity of the Faber splines we overcome the (mixed) smoothness restriction $r<2$ and benefit from higher mixed regularity of the function. As a byproduct we present the solution of Problem 3.13 in the Triebel monograph [46] for the multivariate setting.
