加权多元Paley-Wiener空间在概率框架和平均框架下的Kolmogorov n-宽度
Kolmogorov n-Width of Weighted Multivariate Paley-Wiener Spaces in Probability and Average Settings
摘要: 加权多元Paley-Wiener空间不仅在通讯、信息处理、数据压缩等方面有广泛应用，而且也是逼近定义在 ℝ 上的函数类的重要工具，因而得到广泛的深入研究。本文研究加权多元Paley-Wiener空间在概率框架和平均框架下的逼近特征，特别地，利用离散化的方法估计了在概率框架和平均框架下，加权多元Paley-Wiener空间的Kolmogorov n-宽度的精确渐进阶。
Abstract: Weighted multivariate Paley-Wiener spaces have wide applications in communication, information processing, data compression, and other fields. They are also important tools for approximating classes of functions defined on ℝ , and thus have been extensively studied. This paper studies the approximation characteristics of weighted multivariate Paley-Wiener spaces in probability and average settings. In particular, by using discretization methods, the paper estimates the exact asymptotic order of the Kolmogorov n-width of weighted multivariate Paley-Wiener spaces in the probability and average settings.
文章引用：罗莹. 加权多元Paley-Wiener空间在概率框架和平均框架下的Kolmogorov n-宽度[J]. 应用数学进展, 2025, 14(4): 981-994. https://doi.org/10.12677/aam.2025.144221

参考文献

[1] Cheney, E.W. and Light, W.A. (2009) A Course in Approximation Theory. American Mathematical Soc.
[2] Hyman, C.J. (1956) Theory of Approximation. 1-7.
[3] Kolmogoroff, A. (1936) Über Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse. The Annals of Mathematics, 37, 107-110. [Google Scholar] [CrossRef] 
[4] Li, J. and Wang, H. (2022) Optimal Randomized Quadrature for Weighted Sobolev and Besov Classes with the Jacobi Weight on the Ball. Journal of Complexity, 73, Article 101691. [Google Scholar] [CrossRef] 
[5] Zayed, A.I. (1994) A Sampling Theorem for Signals Bandlimited to a General Domain in Several Dimensions. Journal of Mathematical Analysis and Applications, 187, 196-211. [Google Scholar] [CrossRef] 
[6] Seip, K. (1987) An Irregular Sampling Theorem for Functions Bandlimited in a Generalized Sense. SIAM Journal on Applied Mathematics, 47, 1112-1116. [Google Scholar] [CrossRef] 
[7] Weston, J.D. (1949) XL. A Note on the Theory of Communication. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 40, 449-453. [Google Scholar] [CrossRef] 
[8] Butzer, P.L. (1983) A Survey of the Whittaker-Shannon Sampling Theorem and Some of Its Extensions. Mathematical Research and Exposition, 3, 185-212.
[9] Zayed, A.I. (1992) Kramer’s Sampling Theorem for Multidimensional Signals and Its Relationship with Lagrange-Type Interpolations. Multidimensional Systems and Signal Processing, 3, 323-340. [Google Scholar] [CrossRef] 
[10] Wiener, N. and Paley, R.C. (1934) Fourier Transforms in the Complex Domain. American Mathematical Society, 19, 179-183. [Google Scholar] [CrossRef] 
[11] Li, Y., Chen, G., Xu, Y. and Pan, X. (2024) The Approximation Characteristics of Weighted Band-Limited Function Space. Mathematics, 12, Article 1348. [Google Scholar] [CrossRef] 
[12] Gensun, F. (1996) Whittaker-Kotelnikov-Shannon Sampling Theorem and Aliasing Error. Journal of Approximation Theory, 85, 115-131. [Google Scholar] [CrossRef] 
[13] Bogachev, V. (1998) Gaussian Measures. American Mathematical Society, 62, 361-433. [Google Scholar] [CrossRef] 
[14] Kolmogoroff, A. (1936) Über Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse. The Annals of Mathematics, 37, 107-110. [Google Scholar] [CrossRef] 
[15] Pinkus, A. (1985) N-Widths in Approximation Theory. Springer.
[16] Maĭorov, V.E. (1994) Kolmogorov’s (n, δ)-Widths of Spaces of Smooth Functions. Russian Academy of Sciences. Sbornik Mathematics, 79, 265-279. [Google Scholar] [CrossRef] 
[17] Xu, Y., Chen, G. and Lu, W. (2023) Nonlinear Approximation of Functions by Sets of Finite Pseudo-Dimension in the Probabilistic and Average Case Settings. Analysis and Applications, 21, 1517-1532. [Google Scholar] [CrossRef] 
[18] Fang, G. (2001) Recovery of Band Limited Functions via Cardinal Splines. Science in China Series A: Mathematics, 44, 1126-1131. [Google Scholar] [CrossRef] 
[19] Liu, Y., Li, H. and Li, X. (2023) Approximation Characteristics of Gel’Fand Type in Multivariate Sobolev Spaces with Mixed Derivative Equipped with Gaussian Measure. Axioms, 12, Article 804. [Google Scholar] [CrossRef] 

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