Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives
Izvestiya. Mathematics, Tome 85 (2021) no. 1, pp. 1-23 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, we obtain order estimates for the Kolmogorov widths of the intersection of weighted Sobolev classes with conditions on the first and zeroth derivatives; the weights have power-law form.
Keywords: widths, intersections of function classes, weighted Sobolev classes.
@article{IM2_2021_85_1_a0,
     author = {A. A. Vasil'eva},
     title = {Kolmogorov widths of~intersections of~weighted {Sobolev} classes on an~interval with conditions on the zeroth and first derivatives},
     journal = {Izvestiya. Mathematics},
     pages = {1--23},
     year = {2021},
     volume = {85},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a0/}
}
TY  - JOUR
AU  - A. A. Vasil'eva
TI  - Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives
JO  - Izvestiya. Mathematics
PY  - 2021
SP  - 1
EP  - 23
VL  - 85
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a0/
LA  - en
ID  - IM2_2021_85_1_a0
ER  - 
%0 Journal Article
%A A. A. Vasil'eva
%T Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives
%J Izvestiya. Mathematics
%D 2021
%P 1-23
%V 85
%N 1
%U http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a0/
%G en
%F IM2_2021_85_1_a0
A. A. Vasil'eva. Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives. Izvestiya. Mathematics, Tome 85 (2021) no. 1, pp. 1-23. http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a0/

[1] E. N. Lomakina, V. D. Stepanov, “Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann–Liouville operator”, Siberian Adv. Math., 17:1 (2007), 1–36 | DOI | MR | Zbl

[2] D. E. Edmunds, J. Lang, “Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case”, Math. Nachr., 279:7 (2006), 727–742 | DOI | MR | Zbl

[3] V. N. Konovalov, D. Leviatan, “Kolmogorov and linear widths of weighted Sobolev-type classes on a finite interval”, Anal. Math., 28:4 (2002), 251–278 | DOI | MR | Zbl

[4] J. Lang, “Improved estimates for the approximation numbers of Hardy-type operators”, J. Approx. Theory, 121:1 (2003), 61–70 | DOI | MR | Zbl

[5] M. A. Lifshits, W. Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc., 157, no. 745, Amer. Math. Soc., Providence, RI, 2002, viii+87 pp. | DOI | MR | Zbl

[6] R. Oinarov, “On weighted norm inequalities with three weights”, J. London Math. Soc. (2), 48:1 (1993), 103–116 | DOI | MR | Zbl

[7] V. D. Stepanov, E. P. Ushakova, “On integral operators with variable limits of integration”, Proc. Steklov Inst. Math., 232 (2001), 290–309 | MR | Zbl

[8] P. I. Lizorkin, M. Otelbaev, “Imbedding theorems and compactness for spaces of Sobolev type with weights”, Math. USSR-Sb., 36:3 (1980), 331–349 | DOI | MR | Zbl

[9] P. I. Lizorkin, M. Otelbaev, “Imbedding theorems and compactness for spaces of Sobolev type with weights. II”, Math. USSR-Sb., 40:1 (1981), 51–77 | DOI | MR | Zbl

[10] P. I. Lizorkin, M. O. Otelbaev, “Estimates of approximate numbers of the imbedding operators for spaces of Sobolev type with weights”, Proc. Steklov Inst. Math., 170 (1987), 245–266 | MR | Zbl

[11] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 528 pp. ; North-Holland Math. Library, 18, North-Holland Publishing Co., Amsterdam–New York, 1978, 528 pp. | MR | MR | MR | Zbl | Zbl | Zbl

[12] H. Triebel, “Interpolation properties of $\varepsilon$-entropy and diameters. Geometric characteristics of imbedding for function spaces of Sobolev–Besov type”, Math. USSR-Sb., 27:1 (1975), 23–37 | DOI | MR | Zbl

[13] I. V. Bojkov, “Approximation of some classes of functions by local splines”, Comput. Math. Math. Phys., 38:1 (1998), 21–29 | MR | Zbl

[14] K. T. Mynbaev, M. O. Otelbaev, Vesovye funktsionalnye prostranstva i spektr differentsialnykh operatorov, Nauka, M., 1988, 286 pp. | MR | Zbl

[15] M. S. Aitenova, L. K. Kusainova, “Ob asimptotike raspredeleniya approksimativnykh chisel vlozhenii vesovykh klassov Soboleva. I”, Matem. zhurn., 2:1(3) (2002), 3–9 (electronic) | MR | Zbl

[16] M. S. Aitenova, L. K. Kusainova, “Ob asimptotike raspredeleniya approksimativnykh chisel vlozhenii vesovykh klassov Soboleva. II”, Matem. zhurn., 2:2(4) (2002), 7–14 (electronic) | MR | Zbl

[17] A. A. Vasil'eva, Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain with conditions on the derivatives of order $r$ and zero, 2020, arXiv: 2004.06013v2

[18] V. M. Tikhomirov, Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976, 304 pp. | MR

[19] V. M. Tikhomirov, “Approximation theory”, Analysis. II. Convex analysis and approximation theory, Encyclopaedia Math. Sci., 14, Springer-Verlag, Berlin, 1990, 93–243 | DOI | MR | MR | Zbl | Zbl

[20] A. Pinkus, $n$-widths in approximation theory, Ergeb. Math. Grenzgeb. (3), 7, Springer-Verlag, Berlin, 1985, x+291 pp. | DOI | MR | Zbl

[21] A. Pietsch, “$s$-numbers of operators in Banach spaces”, Studia Math., 51 (1974), 201–223 | DOI | MR | Zbl

[22] M. L. Stesin, “Aleksandrov diameters of finite-dimensional sets and classes of smooth functions”, Soviet Math. Dokl., 16 (1975), 252–256 | MR | Zbl

[23] R. S. Ismagilov, “Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials”, Russian Math. Surveys, 29:3 (1974), 169–186 | DOI | MR | Zbl

[24] E. D. Gluskin, “Norms of random matrices and widths of finite-dimensional sets”, Math. USSR-Sb., 48:1 (1984), 173–182 | DOI | MR | Zbl

[25] B. S. Kašin, “Diameters of some finite-dimensional sets and classes of smooth functions”, Math. USSR-Izv., 11:2 (1977), 317–333 | DOI | MR | Zbl

[26] V. E. Maiorov, “Diskretizatsiya zadachi o poperechnikakh”, UMN, 30:6(186) (1975), 179–180 | MR | Zbl

[27] A. Yu. Garnaev, E. D. Gluskin, “On widths of the Euclidean ball”, Soviet Math. Dokl., 30 (1984), 200–204 | MR | Zbl

[28] B. S. Kashin, “O poperechnikakh oktaedrov”, UMN, 30:4(184) (1975), 251–252 | MR | Zbl

[29] È. M. Galeev, “The Kolmogorov diameter of the intersection of classes of periodic functions and of finite-dimensional sets”, Math. Notes, 29:5 (1981), 382–388 | DOI | MR | Zbl

[30] E. M. Galeev, “Kolmogorov $n$-width of some finite-dimensional sets in a mixed measure”, Math. Notes, 58:1 (1995), 774–778 | DOI | MR | Zbl

[31] È. M. Galeev, “Kolmogorov widths of classes of periodic functions of one and several variables”, Math. USSR-Izv., 36:2 (1991), 435–448 | DOI | MR | Zbl

[32] E. M. Galeev, “Poperechniki funktsionalnykh klassov i konechnomernykh mnozhestv”, Vladikavk. matem. zhurn., 13:2 (2011), 3–14 | MR | Zbl

[33] E. D. Gluskin, “Peresecheniya kuba s oktaedrom plokho approksimiruyutsya podprostranstvami maloi razmernosti”, Priblizhenie funktsii spetsialnymi klassami operatorov, Mezhvuz. sb. nauch. tr., Min. pros. RSFSR, Vologodskii gos. ped. in-t, Vologda, 1987, 35–41 | MR | Zbl

[34] A. D. Izaak, “Kolmogorov widths in finite-dimensional spaces with mixed norms”, Math. Notes, 55:1 (1994), 30–36 | DOI | MR | Zbl

[35] A. D. Izaak, “Widths of Hölder–Nikol'skii classes and finite-dimensional subsets in spaces with mixed norm”, Math. Notes, 59:3 (1996), 328–330 | DOI | DOI | MR | Zbl

[36] 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 | DOI | DOI | MR | Zbl