非光滑多变量函数的近似在奇点附近通常产生低阶近似。大多数先前工作解决了一元函数的问题。本文引入一种方法，用于近似形式为f = g + r_+的非光滑多变量函数，其中g, r ∈ C^{M+1}(R^n)，r_+由r_+(y) = {r(y), r(y) ≥ 0; 0, r(y) < 0}定义。给定散乱（或均匀）数据点X ⊂ R^n，我们研究通过拟插值进行近似。我们设计一个修正项，使得修正后的近似在全域达到全近似阶数。我们还证明修正项是移动最小二乘（MLS）问题的解，因此易于计算且光滑。最后，我们证明所提方法包含对奇点位置的高阶近似。
英文摘要：
Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form $f = g + r_+$ where $g,r \in C^{M+1}(\mathbb{R}^n)$ and the function $r_+$ is defined by \[ r_+(y) = \left\{ \begin{array}{ll} r(y), & r(y) \geq 0 \\ 0, & r(y) < 0 \end{array} \right. \ , \ \forall y \in \mathbb{R}^n \ . \] Given scattered (or uniform) data points $X \subset \mathbb{R}^n$, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.
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On exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.
In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously available literature covered only weights of a simple form commonly known as product weights. In this paper we address the computational aspect of the construction. We develop fast CBC constructio…
Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions $φ_j$ and a sequence of distributions/functions $\widetildeφ_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for $φ_j$ and the compatibility conditions for $φ_j$ and $\widetildeφ_j$. These estimates are given in terms of the Fourier coefficients of approximated functions and provide analogs of some known non-periodic results. Under some additional assumptions error estima…
A least-squares method for sparse low rank approximation of multivariate functions
Ecole Centrale Nantes, Université de Nantes, GeM UMR CNRS 6183（南特中央理工学院，南特大学，GeM UMR CNRS 6183）；Department of Applied Mathematics and Modeling, Airbus Group（空中客车集团应用数学与建模系）
Approximation properties of quasi-projection operators $Q_j(f,φ, \widetildeφ)$ are studied. Such an operator is associated with a function $φ$ satisfying the Strang-Fix conditions and a tempered distribution $\widetildeφ$ such that compatibility conditions with $φ$ hold. Error estimates in the uniform norm are obtained for a wide class of quasi-projection operators defined on the space of uniformly continuous functions and on the anisotropic Besov spaces. Under additional assumptions on $φ$ and $\widetildeφ$, two-sided estimates in terms of realizations of the $K$-functional are also obtained.
Tensor-based multivariate function approximation: methods benchmarking and comparison
ONERA, DTIS, Université de Toulouse（法国图卢兹大学）；Max Planck Institute, CSC Group（马克斯·普朗克研究所）；Department of Electrical and Computer Engineering, Rice University（莱斯大学）
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