多维切比雪夫空间、无限维空间层级与柯尔莫哥洛夫-盖尔范德宽度
Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths
AI总结：
本文引入基于高阶椭圆方程解的多维切比雪夫空间，建立无限维空间层级，并计算多维域上特殊椭球函数集的宽度。
摘要：
近年来，由于柯尔莫哥洛夫宽度（尤其是盖尔范德宽度）理论与新兴的压缩感知领域密切相关，引起了广泛关注。人们认识到宽度能正确反映信号处理中数据的稀疏性。然而，多维函数论中宽度理论的基本问题仍未解决，其进展将对多维信号分析理论中的类似问题产生重大影响。本文有三个主要贡献：1. 解决了寻找切比雪夫系统多维推广的长期问题：我们基于高阶椭圆方程的解引入多维切比雪夫空间，作为一维切比雪夫系统（更确切地说是ECT系统）的推广。2. 在此基础上，我们为定义在多维域上的函数引入了一个新的无限维空间层级；定义了相应的柯尔莫哥洛夫宽度的推广。3. 通过计算定义在多维域上的特殊“椭球”函数集的宽度，推广了柯尔莫哥洛夫的原结果。
英文摘要：
Recently the theory of widths of Kolmogorov (especially of Gelfand widths) has received a great deal of interest due to its close relationship with the newly born area of Compressed Sensing. It has been realized that widths reflect properly the sparsity of the data in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, and their progress will have a major impact over analogous problems in the theory of multidimensional Signal Analysis. The present paper has three major contributions: 1. We solve the longstanding problem of finding multidimensional generalization of the Chebyshev systems: we introduce Multidimensional Chebyshev spaces, based on solutions of higher order elliptic equation, as a generalization of the one-dimensional Chebyshev systems, more precisely of the ECT--systems. 2. Based on that we introduce a new hierarchy of infinite-dimensional spaces for functions defined in multidimensional domains; we define corresponding generalization of Kolmogorov's widths. 3. We generalize the original results of Kolmogorov by computing the widths for special "ellipsoidal" sets of functions defined in multidimensional domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
