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加权多元Paley-Wiener空间在概率框架和平均框架下的Kolmogorov n-宽度
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Abstract:
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
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。
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
, 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.
| [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. https://doi.org/10.2307/1968691 |
| [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. https://doi.org/10.1016/j.jco.2022.101691 |
| [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. https://doi.org/10.1006/jmaa.1994.1352 |
| [6] | Seip, K. (1987) An Irregular Sampling Theorem for Functions Bandlimited in a Generalized Sense. SIAM Journal on Applied Mathematics, 47, 1112-1116. https://doi.org/10.1137/0147073 |
| [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. https://doi.org/10.1080/14786444908521732 |
| [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. https://doi.org/10.1007/bf01940228 |
| [10] | Wiener, N. and Paley, R.C. (1934) Fourier Transforms in the Complex Domain. American Mathematical Society, 19, 179-183. https://doi.org/10.1090/coll/019 |
| [11] | Li, Y., Chen, G., Xu, Y. and Pan, X. (2024) The Approximation Characteristics of Weighted Band-Limited Function Space. Mathematics, 12, Article 1348. https://doi.org/10.3390/math12091348 |
| [12] | Gensun, F. (1996) Whittaker-Kotelnikov-Shannon Sampling Theorem and Aliasing Error. Journal of Approximation Theory, 85, 115-131. https://doi.org/10.1006/jath.1996.0033 |
| [13] | Bogachev, V. (1998) Gaussian Measures. American Mathematical Society, 62, 361-433. https://doi.org/10.1090/surv/062 |
| [14] | Kolmogoroff, A. (1936) Über Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse. The Annals of Mathematics, 37, 107-110. https://doi.org/10.2307/1968691 |
| [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. https://doi.org/10.1070/sm1994v079n02abeh003499 |
| [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. https://doi.org/10.1142/s0219530523500227 |
| [18] | Fang, G. (2001) Recovery of Band Limited Functions via Cardinal Splines. Science in China Series A: Mathematics, 44, 1126-1131. https://doi.org/10.1007/bf02877429 |
| [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. https://doi.org/10.3390/axioms12090804 |
