Abstract:
Precise orders are given for the Kolmogorov and linear widths of the unit ball of the space $l_p^m$ in the metric of $l_q^m$ for $q<\infty$. The determination of the upper estimates is based on approximation by random objects. This method goes back to Kashin (Izv. Akad. Nauk SSSR, Ser. Mat., 1977, vol. 41, p. 334–351). The corresponding lower estimates were obtained in a previous article of the author (Vestn. Leningr. Univ., 1981, № 13, p. 5–10).
Bibliography: 12 titles.
\Bibitem{Glu83}
\by E.~D.~Gluskin
\paper Norms of random matrices and widths of finite-dimensional sets
\jour Math. USSR-Sb.
\yr 1984
\vol 48
\issue 1
\pages 173--182
\mathnet{http://mi.mathnet.ru/eng/sm2114}
\crossref{https://doi.org/10.1070/SM1984v048n01ABEH002667}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=687610}
\zmath{https://zbmath.org/?q=an:0558.46013|0528.46015}
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https://doi.org/10.1070/SM1984v048n01ABEH002667
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This publication is cited in the following 109 articles:
A. A. Vasil'eva, “Estimates for the Kolmogorov widths of an intersection of two balls in a mixed norm”, Sb. Math., 215:1 (2024), 74–89
A. A. Vasil'eva, “Kolmogorov widths of an intersection of a finite family of Sobolev classes”, Izv. Math., 88:1 (2024), 18–42
A.A. Vasil'eva, “Kolmogorov widths of an intersection of a family of balls in a mixed norm”, Journal of Approximation Theory, 2024, 106046
A. A. Vasil'eva, “Kolmogorov widths of a Sobolev class with constraints on derivatives in different metrics”, Sb. Math., 215:11 (2024), 1468–1498
A. A. Vasil'eva, “Kolmogorov widths of anisotropic function classes and finite-dimensional balls”, Eurasian Math. J., 15:3 (2024), 88–93
B. S. Kashin, “A geometric approach to lower bounds for the maximum of Gaussian random processes”, Math. Notes, 116:6 (2024), 1306–1311
A. A. Vasil'eva, “Kolmogorov Widths of the Intersection of Two Finite-Dimensional
Balls in a Mixed Norm”, Math. Notes, 113:4 (2023), 584–586
A. A. Vasileva, “Kolmogorovskie poperechniki peresecheniya dvukh vesovykh klassov Coboleva na otrezke s odinakovoi gladkostyu”, Tr. IMM UrO RAN, 29, no. 4, 2023, 55–63
G. A. Akishev, “Ob otsenkakh lineinykh poperechnikov klassov funktsii mnogikh peremennykh v prostranstve Lorentsa”, Tr. IMM UrO RAN, 28, no. 4, 2022, 23–39
A.A. Vasil'eva, “Kolmogorov widths of intersections of finite-dimensional balls”, Journal of Complexity, 72 (2022), 101649
A. A. Vasil'eva, “Bounds for the Kolmogorov Widths of the Sobolev Weighted Classes with Conditions on the Zero and Highest Derivatives”, Russ. J. Math. Phys., 29:2 (2022), 249
A. A. Vasil'eva, “Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives”, Izv. Math., 85:1 (2021), 1–23
A. A. Vasil'eva, “Kolmogorov widths of the intersection of two finite-dimensional balls”, Russian Math. (Iz. VUZ), 65:7 (2021), 17–23
Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Multi-Dimensional Domain With Conditions on the Derivatives of Order R and Zero”, J. Approx. Theory, 269 (2021), 105602
Yiming Xu, Akil Narayan, Hoang Tran, Clayton G. Webster, “Analysis of the ratio of ℓ1 and ℓ2 norms in compressed sensing”, Applied and Computational Harmonic Analysis, 55 (2021), 486
A. A. Vasil'eva, “Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives”, Russ. J. Math. Phys., 27:4 (2020), 537
A. A. Vasileva, “Kolmogorovskie poperechniki klassov Soboleva na otrezke s ogranicheniyami na variatsiyu”, Tr. IMM UrO RAN, 25, no. 2, 2019, 48–66
Vasil'eva A.A., “Diameters of Sobolev Weight Classes With a “Small” Set of Singularities For Weights”, Russ. J. Math. Phys., 26:4 (2019), 517–543
B. S. Kashin, Yu. V. Malykhin, K. S. Ryutin, “Kolmogorov width and approximate rank”, Proc. Steklov Inst. Math., 303 (2018), 140–153
Qilin Sun, Xiong Dun, Yifan Peng, Wolfgang Heidrich, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018, 273
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