本文给出了基于贪心算法的构造性方法，为具有小混合光滑度的函数类提供了在三角系下$ m $-项逼近的最佳可能误差顺序。
英文摘要：
The paper gives a constructive method, based on greedy algorithms, that provides for the classes of functions with small mixed smoothness the best possible in the sense of order approximation error for the $m$-term approximation with respect to the trigonometric system.
The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier series and as discrete trigonometric Fourier series according to certain systems of functions that are smoothness carriers. It is argued that with approximation and smoothing of trigonometric splines it is expedient to consider as discrete rows, since their differential properties are stored.
Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.
In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical a…
We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where e…
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank…
Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces ${\mathcal S}_{M}$. In particular, in terms of approximations by such methods, constructive characteristics are obtained for classes of functions whose smoothness moduli do not exceed a certain majorant.
We provide a lower bound showing that the $O(1/k)$ convergence rate of the NoLips method (a.k.a. Bregman Gradient) is optimal for the class of functions satisfying the $h$-smoothness assumption. This assumption, also known as relative smoothness, appeared in the recent developments around the Bregman Gradient method, where acceleration remained an open issue. On the way, we show how to constructively obtain the corresponding worst-case functions by extending the computer-assisted performance estimation framework of Drori and Teboulle (Mathematical Programming, 2014) to Bregman first-order meth…
Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^γ$
Hausdorff-Center for Mathematics（豪斯多夫数学中心）；Vietnam National University, Hanoi, Information Technology Institute（越南国家大学河内分校信息技术研究所）；Friedrich-Schiller-University Jena（耶拿弗里德里希·席勒大学）
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of a…
This paper describes an extension of Fourier approximation methods for multivariate functions defined on the torus $\mathbb{T}^d$ to functions in a weighted Hilbert space $L_{2}(\mathbb{R}^d, ω)$ via a multivariate change of variables $ψ:\left(-\frac{1}{2},\frac{1}{2}\right)^d\to\mathbb{R}^d$. We establish sufficient conditions on $ψ$ and $ω$ such that the composition of a function in such a weighted Hilbert space with $ψ$ yields a function in the Sobolev space $H_{\mathrm{mix}}^{m}(\mathbb{T}^d)$ of functions on the torus with mixed smoothness of natural order $m \in \mathbb{N}_{0}$. In this …
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $\mathbb{X}$. First, we generalize a result of Temlyakov to cover situations in which the modulus of smoothness and the so called A3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces $\mathbb{X}=L^p(\log L)^α$, with $1<p<\infty$ and $α\in\mathbb{R}$, and show that, when the Haar system is used, then optimal recovery of $N$-sparse signals occurs when the number of iterations is $ϕ(N)=O(N^{\max\{1…
University of Chinese Academy of Sciences（中国科学院大学）；National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences（中国科学院国家数学与交叉科学中心）
Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. …
In this paper we propose a tool for high-dimensional approximation based on trigonometric polynomials where we allow only low-dimensional interactions of variables. In a general high-dimensional setting, it is already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. This requires black-box access to the function, i.e., the ability to evaluate it at any point. Here, we focus on scattered data points and grouped frequency index sets along the dimensions. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, for high-dimensio…
We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$ error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces (RHKS) using approximants that are trigonometric polynomials computed from sampling values. The used sampling schemes are suitable sets of rank-1 lattices that can be constructed in an extremely efficient way. Furthermore, the structure of the sampling schemes allows for fast Fourier transform (FFT) algorithms. We present and discuss one FFT algorithm and analyze the worst case $L_\infty$ error for this…
We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow any number of points, including powers of $2$, thus providing the fundamental theory for construction of embedded lattice sequences. Our results are constructive in that we provide a component-by-component algorithm which constructs a suitable generating vector for a given number of points or even a range of numbers of points. It does so without needing to co…
