Abstract
We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.
Highlights
Рассматривается задача про порядок приближения дваджды непрерывно дифференцируемых функций многих переменных кусочно-постоянными на выпуклых разбиениях
We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants
We show that the saturation order of piecewise constant approximation in Lp norm on convex partitions with N cells is N −2/(d+1), where d is the number of variables
Summary
N −1/d can be achieved for a smooth non-constant function on “isotropic” partitions (see [2]). It was shown in [2, 3] that on a wider set of all convex partitions DN essentially better order of approximation is possible. As N → ∞, which almost doubles the approximation order in comparison to N −1/d when d is large enough. This improvement in order is achieved on “anisotropic” sequence of partitions. ∇f (x) is the gradient of function f at the point x
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