最佳恢复和容量估计

IF 1.8 2区数学 Q1 MATHEMATICS

Journal of Complexity Pub Date : 2023-06-17 DOI:10.1016/j.jco.2023.101780

Alexander Kushpel

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引用次数: 0

摘要

研究了由概率空间上的正交系统诱导的Rn中凸原点对称体的截面体积。该方法基于John-Löwner椭球体的体积估计和各自系统引起的规范期望。得到的估计使我们能够建立截面半径的下界，从而给出Gelfand宽度(或线性宽度)的下界。作为应用，我们给出了一种新的求乘子算子的Gelfand宽度和Kolmogorov宽度的方法。特别地，我们在两点齐次空间上，即当1<q≤p≤∞的困难情况下，建立了Lq上标准Sobolev类Wpγ， γ>0的宽度的尖锐阶。

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Optimal recovery and volume estimates

We study volumes of sections of convex origin-symmetric bodies in $R^{n}$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes $W_{p}^{γ}$ , $γ > 0$ in $L_{q}$ on two-point homogeneous spaces in the difficult case, i.e. if $1 < q \leq p \leq \infty$ .

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.