A general theorem on the existance of lipshitz retractions on certain surfaces was proved.<br> As a corollary, a lipshitz $\varepsilon$-selection on the set of generalised rational functions was constructed. Several results, concerning the problem of existance of continuous selections from the operator of generalised rational approximation in $L_p$, $0<p\le\infty$ spaces were obtained. It was proved that pseudodimension of any set is greater or equal than its topological dimension.
Main publications:
Lipshitsevost retraktsii i operator obobschennogo ratsionalnogo priblizheniya // Fundament. i prikl. matem., 2000, # 4, s. 1205–1220.
Ravnomernaya nepreryvnost obobschennykh ratsionalnykh priblizhenii // Matematicheskie zametki, 2002, t. 71, # 2, s. 261–270.
Otsenka snizu psevdorazmernosti cherez topologicheskuyu // Matematicheskie zametki, 2001, t. 70, # 1, s. 155–156.
A. A. Vasil'eva, A. V. Gorshkov, M. P. Zapletin, L. V. Lokutsievskiy, G. G. Magaril-Il'yaev, K. Yu. Osipenko, K. S. Ryutin, A. V. Fursikov, “Main recent research results of the staff of the Chair of General Problems of Control”, Moscow University Mathematics Bulletin, 79:6 (2024), 351–359
5.
A. R. Alimov, K. S. Ryutin, I. G. Tsar'kov, “Existence, uniqueness, and stability of best and near-best approximations”, Russian Math. Surveys, 78:3 (2023), 399–442
6.
Yuri Malykhin, Konstantin Ryutin, Tatyana Zaitseva, “Recovery of regular ridge functions on the ball”, Constr. Approx., 56 (2022), 687–708 , arXiv: 2102.13203
T. I. Zaitseva, Yu. V. Malykhin, K. S. Ryutin, “On Recovery of Regular Ridge Functions”, Math. Notes, 109:2 (2021), 307–311
8.
A. S. Rubtsova, K. S. Ryutin, V. N. Temlyakov, “On the fixed volume discrepancy of the Korobov point sets”, Sb. Math., 212:8 (2021), 1180–1192
9.
K. S. Ryutin, “Integer Sampling Matrices with Small Entries Ensuring Vector Recovery”, Math. Notes, 107:2 (2020), 363–366
10.
K. S. Ryutin, “Recovery of sparse integer vectors from linear measurements”, Russian Math. Surveys, 74:6 (2019), 1129–1131
11.
B. S. Kashin, Yu. V. Malykhin, K. S. Ryutin, “Kolmogorov width and approximate rank”, Proc. Steklov Inst. Math., 303 (2018), 140–153
12.
Yu. V. Malykhin, K. S. Ryutin, “The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric”, Math. Notes, 101:1 (2017), 94–99
13.
Yu. V. Malykhin, K. S. Ryutin, “Concentration of the $L_1$-norm of trigonometric polynomials and entire functions”, Sb. Math., 205:11 (2014), 1620–1649
14.
Yu. V. Malykhin, K. S. Ryutin, “On the concentration of measure and the $L^1$-norm”, J. Approx. Theory, 175 (2013), 77–82
15.
S. V. Konyagin, Yu. V. Malykhin, C. S. Rjutin, “On Exact Recovery of Sparse Vectors from Linear Measurements”, Math. Notes, 94:1 (2013), 107–114
16.
C. S. Rjutin, “On Uniformly Continuous Almost Best Generalized Rational Approximation Operators”, Math. Notes, 87:1 (2010), 141–145
17.
C. S. Rjutin, “Continuity of Operators of Generalized Rational Approximation in the Space $L_1[0;1]$”, Math. Notes, 73:1 (2003), 142–147
18.
C. S. Rjutin, “Uniform Continuity of Generalized Rational Approximations”, Math. Notes, 71:2 (2002), 236–244
19.
C. S. Rjutin, “A Lower Bound for Pseudodimension in Terms of Topological Dimension”, Math. Notes, 70:1 (2001), 143–144
20.
C. S. Rjutin, “Lipshitz retractions and the operator of generalized rational approximation”, Fundam. Prikl. Mat., 6:4 (2000), 1205–1220
21.
Teoriya priblizhenii, funktsionalnyi analiz i prilozheniya, Sbornik statei. K 70-letiyu akademika Borisa Sergeevicha Kashina, Trudy MIAN, 319, ed. S. V. Konyagin, M. I. Dyachenko, V. N. Temlyakov, M. R. Gabdullin, K. S. Ryutin, MIAN, M., 2022 , 328 pp.
22.
A. R. Alimov, A. A. Vasil'eva, S. V. Konyagin, A. S. Kochurov, A. I. Kozko, T. P. Lukashenko, K. S. Ryutin, V. M. Tikhomirov, A. V. Fursikov, V. N. Chubarikov, “Igor Germanovich Tsarkov (to 60th anniversary)”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2020, no. 4, 70–71
23.
Garmonicheskii analiz, teoriya priblizhenii i teoriya chisel, Sbornik statei. K 60-letiyu so dnya rozhdeniya akademika Sergeya Vladimirovicha Konyagina, Trudy MIAN, 303, ed. B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, I. D. Shkredov, K. S. Ryutin, MAIK «Nauka/Interperiodika», M., 2018 , 287 pp.
24.
B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, K. S. Ryutin, I. D. Shkredov, “Sergei Vladimirovich Konyagin turns 60”, Proc. Steklov Inst. Math., 303 (2018), 1–9
25.
S. V. Konyagin, Yu. V. Malykhin, K. S. Ryutin, “Letter to the Editor”, Math. Notes, 96:3 (2014), 462
An extremal problem for integer sparse recovery S. V. Konyagin, K. S. Rjutin XVI International Conference «Algebra, Number Theory and Discrete Geometry: modern problems, applications and problems of history» dedicated to the 80th anniversary of the birth of Professor Michel Desa May 13, 2019 15:00
On the concentration of $L_1$ norm Yu. V. Malykhin, K. S. Ryutin International conference "Nonlinear Approximations and Applications" dedicated to the 60th birthday of Professor V. N. Temlyakov November 1, 2013 17:00
Approximation Theory, Functional Analysis, and Applications, Collected papers. On the occasion of the 70th birthday of Academician Boris Sergeevich Kashin, Trudy Mat. Inst. Steklova, 319, ed. S. V. Konyagin, M. I. Dyachenko, V. N. Temlyakov, M. R. Gabdullin, K. S. Ryutin, 2022, 328 с. http://mi.mathnet.ru/book1903
Harmonic analysis, approximation theory, and number theory, Collected papers. Dedicated to Academician Sergei Vladimirovich Konyagin on the occasion of his 60th birthday, Trudy Mat. Inst. Steklova, 303, ed. B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, I. D. Shkredov, K. S. Ryutin, 2018, 287 с. http://mi.mathnet.ru/book1734
