Kolmogorov widths and approximation numbers of Sobolev classes with singular weights

Vasil′eva, A.

doi:10.1090/S1061-0022-2012-01229-X

Kolmogorov widths and approximation numbers of Sobolev classes with singular weights
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by A. A. Vasil′eva
Translated by: the author

St. Petersburg Math. J. 24 (2013), 1-27

DOI: https://doi.org/10.1090/S1061-0022-2012-01229-X

Published electronically: November 15, 2012

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Abstract:

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.

References

Stefan Heinrich, On the relation between linear $n$-widths and approximation numbers, J. Approx. Theory 58 (1989), no. 3, 315–333. MR 1012680, DOI 10.1016/0021-9045(89)90032-4
V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Russian Math. Surveys 15 (1960), no. 3, 75–111. MR 0117489, DOI 10.1070/RM1960v015n03ABEH004093
V. M. Tihomirov and S. B. Babadžanov, Diameters of a function class in an $L_{p}$-space $(p\geq 1)$, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 11 (1967), no. 2, 24–30 (Russian, with Uzbek summary). MR 0209747
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. MR 1099516, DOI 10.1070/SM1992v071n02ABEH001404
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). MR 0407509
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). MR 0481792
Albrecht Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. MR 361883, DOI 10.4064/sm-51-3-201-223
M. I. Stesin, Aleksandrov diameters of finite-dimensional sets and of classes of smooth functions, Dokl. Akad. Nauk SSSR 220 (1975), 1278–1281 (Russian). MR 0617942
V. E. Maĭorov, Discretization of the problem of diameters, Uspehi Mat. Nauk 30 (1975), no. 6(186), 179–180 (Russian). MR 0402349
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). MR 0291774
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). MR 687610
V. M. Tikhomirov, Nekotorye voprosy teorii priblizheniĭ, Izdat. Moskov. Univ., Moscow, 1976 (Russian). MR 0487161
V. M. Tikhomirov, Approximation theory, Current problems in mathematics. Fundamental directions, Vol. 14 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 103–260, 272 (Russian). MR 915774
Allan Pinkus, $n$-widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. MR 774404, DOI 10.1007/978-3-642-69894-1
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. MR 1895252, DOI 10.1090/memo/0745
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. MR 2226408, DOI 10.1002/mana.200510389
J. Lang, Improved estimates for the approximation numbers of Hardy-type operators, J. Approx. Theory 121 (2003), no. 1, 61–70. MR 1962996, DOI 10.1016/S0021-9045(02)00043-6
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. MR 1974800
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). MR 2251331
A. A. Vasil’eva, Estimates for the widths of weighted Sobolev classes, Sb. Math. 201 (2010), no. 7-8, 947–984. MR 2907813, DOI 10.1070/SM2010v201n07ABEH004098
A. A. Vasil′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. MR 2587811, DOI 10.1134/S1061920809040098
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. MR 995015, DOI 10.1007/BF01054210
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. MR 1072699, DOI 10.1070/IM1991v036n03ABEH002039
—, Two-weighted estimates for Riemann–Liouville integrals, Rep. no. 39, Ceskoslov. Akad. Věd. Mat. Ústav., Prague, 1988, pp. 1–28.
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. MR 1322312
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. MR 1277757, DOI 10.1112/jlms/50.1.105
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). MR 1097892
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. MR 2408040, DOI 10.5209/rev_{R}EMA.2008.v21.n1.16447
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. MR 2827745, DOI 10.1155/2011/928962
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454