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Kolmogorov widths and approximation numbers of Sobolev classes with singular weights. (English. Russian original) Zbl 1275.41033

St. Petersbg. Math. J. 24, No. 1, 1-27 (2013); translation from Algebra Anal. 24, No. 1, 3-39 (2012).
Summary: The Kolmogorov widths of weighted Sobolev classes in weighted \(L_q\)-spaces and the approximation numbers of the corresponding embedding operators are estimated. The case where the weights affect the asymptotics is considered.

MSC:

41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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[1] Stefan Heinrich, On the relation between linear \?-widths and approximation numbers, J. Approx. Theory 58 (1989), no. 3, 315 – 333. · Zbl 0699.41022 · doi:10.1016/0021-9045(89)90032-4
[2] V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Russian Math. Surveys 15 (1960), no. 3, 75 – 111. · Zbl 0097.09103 · doi:10.1070/RM1960v015n03ABEH004093
[3] V. M. Tihomirov and S. B. Babadžanov, Diameters of a function class in an \?_{\?}-space (\?\ge 1), Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 11 (1967), no. 2, 24 – 30 (Russian, with Uzbek summary).
[4] A. P. Buslaev and V. M. Tikhomirov, The spectra of nonlinear differential equations and widths of Sobolev classes, Mat. Sb. 181 (1990), no. 12, 1587 – 1606 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 2, 427 – 446. · Zbl 0718.34113
[5] R. S. Ismagilov, Diameters of sets in normed linear spaces, and the approximation of functions by trigonometric polynomials, Uspehi Mat. Nauk 29 (1974), no. 3(177), 161 – 178 (Russian). · Zbl 0303.41039
[6] B. S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 334 – 351, 478 (Russian).
[7] Albrecht Pietsch, \?-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201 – 223. · Zbl 0294.47018
[8] M. I. Stesin, Aleksandrov diameters of finite-dimensional sets and of classes of smooth functions, Dokl. Akad. Nauk SSSR 220 (1975), 1278 – 1281 (Russian). · Zbl 0333.46012
[9] V. E. Maĭorov, Discretization of the problem of diameters, Uspehi Mat. Nauk 30 (1975), no. 6(186), 179 – 180 (Russian).
[10] Ju. I. Makovoz, A certain method of obtaining lower estimates for diameters of sets in Banach spaces, Mat. Sb. (N.S.) 87(129) (1972), 136 – 142 (Russian).
[11] E. D. Gluskin, Norms of random matrices and diameters of finite-dimensional sets, Mat. Sb. (N.S.) 120(162) (1983), no. 2, 180 – 189, 286 (Russian). · Zbl 0528.46015
[12] Некоторые вопросы теории приближений., Издат. Москов. Унив., Мосцощ, 1976 (Руссиан).
[13] Современные проблемы математики. Фундаментал\(^{\приме}\)ные направления, Том 14, Акад. Наук СССР, Всесоюз. Инст. Научн. и Техн. Информ., Мосцощ, 1987 [ МР0915772 (89ф:41001)]; Транслатион бы Д. Нещтон.
[14] Allan Pinkus, \?-widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. · Zbl 0551.41001
[15] Mikhail A. Lifshits and Werner Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc. 157 (2002), no. 745, viii+87. · Zbl 0999.47034 · doi:10.1090/memo/0745
[16] D. E. Edmunds and J. Lang, Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case, Math. Nachr. 279 (2006), no. 7, 727 – 742. · Zbl 1102.47035 · doi:10.1002/mana.200510389
[17] J. Lang, Improved estimates for the approximation numbers of Hardy-type operators, J. Approx. Theory 121 (2003), no. 1, 61 – 70. · Zbl 1035.47009 · doi:10.1016/S0021-9045(02)00043-6
[18] Elena N. Lomakina and Valdimir D. Stepanov, On asymptotic behaviour of the approximation numbers and estimates of Schatten-von Neumann norms of the Hardy-type integral operators, Function spaces and applications (Delhi, 1997) Narosa, New Delhi, 2000, pp. 153 – 187. · Zbl 0998.47033
[19] E. N. Lomakina and V. D. Stepanov, Asymptotic estimates for the approximation and entropy numbers of the one-weight Riemann-Liouville operator, Mat. Tr. 9 (2006), no. 1, 52 – 100 (Russian, with Russian summary). · Zbl 1249.47037
[20] A. A. Vasil’eva, Estimates for the widths of weighted Sobolev classes, Sb. Math. 201 (2010), no. 7-8, 947 – 984. · Zbl 1205.41019 · doi:10.1070/SM2010v201n07ABEH004098
[21] A. A. Vasil\(^{\prime}\)eva, Criterion for the existence of a continuous embedding of a weighted Sobolev class on a closed interval and on a semiaxis, Russ. J. Math. Phys. 16 (2009), no. 4, 543 – 562. · Zbl 1192.46033 · doi:10.1134/S1061920809040098
[22] È. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Sibirsk. Mat. Zh. 30 (1989), no. 1, 13 – 22 (Russian); English transl., Siberian Math. J. 30 (1989), no. 1, 8 – 16. · Zbl 0729.42007 · doi:10.1007/BF01054210
[23] V. D. Stepanov, Two-weight estimates for Riemann-Liouville integrals, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 645 – 656 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 669 – 681. · Zbl 0705.26015
[24] -, Two-weighted estimates for Riemann-Liouville integrals, Rep. no. 39, Ceskoslov. Akad. Věd. Mat. Ústav., Prague, 1988, pp. 1-28.
[25] Vladimir D. Stepanov, Weighted norm inequalities for integral operators and related topics, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994) Prometheus, Prague, 1994, pp. 139 – 175. · Zbl 0866.47025
[26] Vladimir D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, J. London Math. Soc. (2) 50 (1994), no. 1, 105 – 120. · Zbl 0837.26012 · doi:10.1112/jlms/50.1.105
[27] A. Kufner and G. P. Kheĭnig, The Hardy inequality for higher-order derivatives, Trudy Mat. Inst. Steklov. 192 (1990), 105 – 113 (Russian). Translated in Proc. Steklov Inst. Math. 1992, no. 3, 113 – 121; Differential equations and function spaces (Russian). · Zbl 0716.26008
[28] Dorothee D. Haroske and Leszek Skrzypczak, Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights. I, Rev. Mat. Complut. 21 (2008), no. 1, 135 – 177. · Zbl 1202.46039 · doi:10.5209/rev_REMA.2008.v21.n1.16447
[29] Dorothee D. Haroske and Leszek Skrzypczak, Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases, J. Funct. Spaces Appl. 9 (2011), no. 2, 129 – 178. · Zbl 1253.46043 · doi:10.1155/2011/928962
[30] Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. · Zbl 0698.46008
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