Enviar artigo por via de e-mail Some lower bounds for optimal sampling recovery of functions with mixed smoothness
- Autores: Gasnikov A.V.1,2,3, Temlyakov V.N.4,2,5,6
-
Afiliações:
- Ivannikov Institute for System Programming of the Russian Academy of Science, Moscow, Russia
- Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
- Innopolis University, Innopolis, Russia
- University of South Carolina, Columbia, SC, USA
- Lomonosov Moscow State University, Moscow, Russia
- Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
- Edição: Volume 216, Nº 11 (2025)
- Páginas: 90-107
- Seção: Articles
- URL: https://ogarev-online.ru/0368-8666/article/view/351336
- DOI: https://doi.org/10.4213/sm10250
- ID: 351336
Citar
Acesso aberto
Acesso está concedido
Somente assinantes
Resumo
Recently there was a substantial progress in the problem of sampling recovery on function classes with mixed smoothness. It was mostly done by proving new and sometimes optimal upper bounds for both linear sampling recovery and nonlinear sampling recovery. In this paper we address the problem of lower bounds for the optimal rates of nonlinear sampling recovery. In the case of linear recovery one can use the well-developed theory of estimating the Kolmogorov and linear widths to establish some lower bounds for the optimal rates. In the case of nonlinear recovery we cannot use the above approach. It seems like the only technique which is available now is based on some simple observations. We demonstrate how these observations can be used.
Palavras-chave
Sobre autores
Alexander Gasnikov
Ivannikov Institute for System Programming of the Russian Academy of Science, Moscow, Russia; Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; Innopolis University, Innopolis, Russia
Email: gasnikov@yandex.ru
ORCID ID: 0000-0002-7386-039X
Scopus Author ID: 15762551000
Researcher ID: L-6371-2013
Doctor of physico-mathematical sciences, Associate professor
Vladimir Temlyakov
University of South Carolina, Columbia, SC, USA; Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; Lomonosov Moscow State University, Moscow, Russia; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
Email: temlyakovv@gmail.com
Doctor of physico-mathematical sciences, Professor
Bibliografia
- M. Dolbeault, D. Krieg, M. Ullrich, “A sharp upper bound for sampling numbers in $L_2$”, Appl. Comput. Harmon. Anal., 63 (2023), 113–134
- F. Dai, V. Temlyakov, Universal discretization and sparse sampling recovery
- F. Dai, V. Temlyakov, “Random points are good for universal discretization”, J. Math. Anal. Appl., 529:1 (2024), 127570, 28 pp.
- F. Dai, V. Temlyakov, Lebesgue-type inequalities in sparse sampling recovery
- D. Dũng, V. Temlyakov, T. Ullrich, Hyperbolic cross approximation, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2018, xi+218 pp.
- T. Jahn, T. Ullrich, F. Voigtlaender, “Sampling numbers of smoothness classes via $ell^1$-minimization”, J. Complexity, 79 (2023), 101786, 35 pp.
- E. D. Kosov, V. N. Temlyakov, Bounds for the sampling discretization error and their applications to universal sampling discretization
- D. Krieg, M. Ullrich, “Function values are enough for $L_2$-approximation”, Found. Comput. Math., 21:4 (2021), 1141–1151
- D. Krieg, M. Ullrich, “Function values are enough for $L_2$-approximation: Part II”, J. Complexity, 66 (2021), 101569, 14 pp.
- N. Nagel, M. Schäfer, T. Ullrich, “A new upper bound for sampling numbers”, Found. Comput. Math., 22:2 (2022), 445–468
- V. N. Temlyakov, “On approximate recovery of functions with bounded mixed derivative”, J. Complexity, 9:1 (1993), 41–59
- V. Temlyakov, Multivariate approximation, Cambridge Monogr. Appl. Comput. Math., 32, Cambridge Univ. Press, Cambridge, 2018, xvi+534 pp.
- V. Temlyakov, “On optimal recovery in $L_2$”, J. Complexity, 65 (2021), 101545, 11 pp.
- V. Temlyakov, Sparse sampling recovery by greedy algorithms
- V. Temlyakov, Sparse sampling recovery in integral norms on some function classes
- V. Temlyakov, T. Ullrich, “Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness”, J. Complexity, 67 (2021), 101575, 19 pp.
- J. F. Traub, G. W. Wasilkowski, H. Wozniakowski, Information-based complexity, Comput. Sci. Sci. Comput., Academic Press, Inc., Boston, MA, 1988, xiv+523 pp.
Arquivos suplementares
