×

Bounds for the Kolmogorov widths of the Sobolev weighted classes with conditions on the zero and highest derivatives. (English) Zbl 1500.46029

Summary: In this paper, order estimates for the Kolmogorov widths of sets defined by restrictions on the norm in the Sobolev weighted space \(W^r_{p_1}\) and the weighted space \(L_{p_0}\) are obtained.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aitenova, M. S.; Kusainova, L. K., On the Asymptotics of the Distribution of Approximation Numbers of Embeddings of Weighted Sobolev Classes. I, Mat. Zh., 2, 1, 3-9 (2002) · Zbl 1134.46307
[2] Aitenova, M. S.; Kusainova, L. K., On the Asymptotics of the Distribution of Approximation Numbers of Embeddings of Weighted Sobolev Classes. II, Mat. Zh., 2, 2, 7-14 (2002) · Zbl 1134.46308
[3] Boykov, I. V., Approximation of Some Classes of Functions by Local Splines, Comput. Math. Math. Phys., 38, 1, 21-29 (1998) · Zbl 0954.41017
[4] Boykov, I. V.; Ryazantsev, V. A., On the Optimal Approximation of Geophysical Fields, Numer. Anal. Appl., 14, 1, 13-29 (2021)
[5] Bricchi, M., Existence and Properties of \(h\)-Sets, Georgian Math. J., 9, 1, 13-32 (2002) · Zbl 1040.28008
[6] Galeev, E. M., The Kolmogorov Diameter of the Intersection of Classes of Periodic Functions and of Finite-Dimensional Sets, Math. Notes, 29, 5, 382-388 (1981) · Zbl 0507.41020
[7] Galeev, E. M., Widths of Functional Classes and Finite-Dimensional Sets, Vladikavkaz. Mat. Zh., 13, 2, 3-14 (2011) · Zbl 1238.46025
[8] Garnaev, A. Yu.; Gluskin, E. D., On Widths of the Euclidean Ball, Dokl. Akad. Nauk SSSR, 277, 5, 1048-1052 [Sov. Math. Dokl., 30 (1984), 200 (1984) · Zbl 0588.41022
[9] Gluskin, E. D., Norms of Random Matrices and Diameters of Finite-Dimensional Sets, Math. USSR-Sb., 48, 1, 173-182 (1984) · Zbl 0558.46013
[10] Kashin, B. S., The Diameters of Octahedra, Usp. Mat. Nauk, 30, 4, 251-252 (1975) · Zbl 0311.46014
[11] Kashin, B. S., The Widths of Certain Finite-Dimensional Sets and Classes of Smooth Functions, Math. USSR-Izv., 11, 2, 317-333 (1977) · Zbl 0378.46027
[12] Kolmogorov, A. N.; Petrov, A. A.; Smirnov, Yu. M., A Formula of Gauss in the Theory of the Method of Least Squares, Izvestiya Akad. Nauk SSSR. Ser. Mat., 11, 561-566 (1947) · Zbl 0029.40501
[13] Lizorkin, P. I.; Otelbaev, M., Imbedding Theorems and Compactness for Spaces of Sobolev Type with Weights, Math. USSR-Sb., 36, 3, 331-349 (1980) · Zbl 0432.46029
[14] Lizorkin, P. I.; Otelbaev, M., Imbedding Theorems and Compactness for Spaces of Sobolev Type with Weights. II, Math. USSR-Sb., 40, 1, 51-77 (1981) · Zbl 0465.46031
[15] Lizorkin, P. I.; Otelbaev, M., Estimates of Approximate Numbers of the Imbedding Operators for Spaces of Sobolev Type with Weights, Proc. Steklov Inst. Math., 170, 245-266 (1987) · Zbl 0616.46035
[16] Mynbaev, K.; Otelbaev, M., Weighted Function Spaces and the Spectrum of Differential Operators (1988), Moscow: Nauka, Moscow · Zbl 0651.46037
[17] Oinarov, R., On Weighted Norm Inequalities with Three Weights, J. London Math. Soc. (2), 48, 103-116 (1993) · Zbl 0811.26008
[18] Pinkus, A., \(n\)-Widths in Approximation Theory (1985), Berlin: Springer, Berlin · Zbl 0551.41001
[19] Pietsch, A., \(s\)-Numbers of Operators in Banach Space, Studia Math., 51, 201-223 (1974) · Zbl 0294.47018
[20] Reshetnyak, Yu. G., Integral Representations of Differentiable Functions in Domains with a Nonsmooth Boundary, Sibirsk. Mat. Zh., 21, 6, 108-116 (1980) · Zbl 0455.35025
[21] Reshetnyak, Yu. G., A Remark on Integral Representations of Differentiable Functions of Several Variables, Sibirsk. Mat. Zh., 25, 5, 198-200 (1984) · Zbl 0597.26018
[22] Stechkin, S. B., On the Best Approximations of Given Classes of Functions by Arbitrary Polynomials, Uspekhi Mat. Nauk, 9, 1-59, 133-134 (1954) · Zbl 0055.06001
[23] Stepanov, V. D.; Ushakova, E. P., On Integral Operators with Variable Limits of Integration, Proc. Steklov Inst. Math., 232, 290-309 (2001) · Zbl 1007.46036
[24] Stesin, M. I., Aleksandrov Diameters of Finite-Dimensional Sets and of Classes of Smooth Functions, Dokl. Akad. Nauk SSSR, 220, 6, 1278-1281 [Soviet Math. Dokl.] (1975) · Zbl 0333.46012
[25] Tikhomirov, V. M., Theory of Approximations, Current problems in mathematics. Fundamental directions, 14, 0 (0000) · Zbl 0931.41017
[26] Triebel, H., Interpolation Theory. Function Spaces. Differential Operators (1980), Moscow: Mir, Moscow · Zbl 0531.46025
[27] Vasil’eva, A. A., Widths of Weighted Sobolev Classes on a John Domain, Proc. Steklov Inst. Math., 280, 91-119 (2013) · Zbl 1296.46030
[28] Vasil’eva, A. A., Widths of Function Classes on Sets With Tree-Like Structure, J. Approx. Theory, 192, 19-59 (2015) · Zbl 1312.41035
[29] Vasil’eva, A. A., Kolmogorov Widths of the Intersection of Two Finite-Dimensional Balls, Russ. Math., 65, 7, 17-23 (2021) · Zbl 1479.41018
[30] Vasil’eva, A. A., Kolmogorov Widths of Weighted Sobolev Classes on a Multi-Dimensional Domain with Conditions on the Derivatives of Order \(r\) and Zero, J. Approx. Theory, 269, 0 (2021) · Zbl 1471.41013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.