加权带有限函数空间在一致框架下的熵数

Entropy Numbers of Weighted Band-Limited Function Space in Uniform Setting

摘要: 熵数作为衡量函数空间复杂性的核心工具，在人工智能、控制论和科学计算等领域具有重要意义。本文基于已有研究，构建了加权带有限函数空间，采用离散化方法，探讨了加权带有限函数空间在一致框架下的熵数问题，并进一步估计出一致框架下熵数的精确渐近阶，即设 $1<p,q<\infty ,r>max\left\{0,\frac{1}{q}-\frac{1}{p}\right\},n=0,1,2,\cdots$ ，则${\epsilon }_n\left(B_{\sigma ,p}^{\omega }\left(ℝ\right),B_{\sigma ,q}\left(ℝ\right)\right)\asymp n^{-\left(r-\frac{1}{q}+\frac{1}{p}\right)}$ 。其中$B_{\sigma ,p}^{\omega }\left(ℝ\right)$ 是权为$\omega ={\left\{{\left|k\right|}^r\right\}}_{k\in {\mathbb{Z}}_0}$ 的加权带有限函数空间。

Abstract: Entropy numbers, as a core tool for measuring the complexity of function spaces, play a significant role in fields such as artificial intelligence, cybernetics, and scientific computing. Based on existing research, this study constructs weighted band-limited function spaces and employs discretization methods to investigate the entropy numbers of these spaces in uniform setting. Furthermore, the exact asymptotic order of the classical entropy number of the weighted band-limited function spaces is estimated, specifically: If $1<p,q<\infty ,r>max\left\{0,\frac{1}{q}-\frac{1}{p}\right\},n=0,1,2,\cdots$ , then ${\epsilon }_n\left(B_{\sigma ,p}^{\omega }\left(ℝ\right),B_{\sigma ,q}\left(ℝ\right)\right)\asymp n^{-\left(r-\frac{1}{q}+\frac{1}{p}\right)}.$ Where $B_{\sigma ,p}^{\omega }\left(ℝ\right)$ is the weighted band-limited function space characterized by weight $\omega ={\left\{{\left|k\right|}^r\right\}}_{k\in {\mathbb{Z}}_0}$ .