Vasil’eva, Anastasia A. Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives. (English. Russian original) Zbl 1460.41014 Izv. Math. 85, No. 1, 1-23 (2021); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 85, No. 1, 3-26 (2021). Suppose that \(1\leq p_0,p_1,q\leq\infty\), and let \(g(t) = t^{\beta}\), \(w(t) = t^{-\sigma}\) for \(t>0\), where \[ \beta > 1 + \frac{1}{q} - \frac{1}{p_1}, \quad \beta + \sigma > 1 + \frac{1}{p_0} - \frac{1}{p_1}. \] In the paper, the author obtains order estimates for the Kolmogorov widths of the classes \[ M_{p_0,p_1,g,w} :=\{f\in AC[0,1]\,: \ \|gf'\|_{L_{p_1}[0,1]}\leq 1, \ \|wf\|_{L_{p_0}[0,1]}\leq 1 \} \] in a space \(L_q[0,1]\). The main result without proof is published in the note [A. A. Vasil’eva, Math. Notes 107, No. 3, 522–524 (2020; Zbl 1473.41007); translation from Mat. Zametki 107, No. 3, 470–472 (2020)]. Reviewer: Yuri A. Farkov (Moskva) MSC: 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy Keywords:widths; intersections of function classes; weighted Sobolev classes Citations:Zbl 1473.41007 PDFBibTeX XMLCite \textit{A. A. Vasil'eva}, Izv. Math. 85, No. 1, 1--23 (2021; Zbl 1460.41014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 85, No. 1, 3--26 (2021) Full Text: DOI References: [1] Lomakina E. N. and Stepanov V. D. 2006 Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann-Liouville operator Matem. Tr.9 52-100 · Zbl 1249.47037 [2] Edmunds D. E. and Lang J. 2006 Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case Math. Nachr.279 727-742 · Zbl 1102.47035 [3] Konovalov V. N. and Leviatan D. 2002 Kolmogorov and linear widths of weighted Sobolev-type classes on a finite interval Anal. Math.28 251-278 · Zbl 1017.46026 [4] Lang J. 2003 Improved estimates for the approximation numbers of Hardy-type operators J. Approx. Theory121 61-70 · Zbl 1035.47009 [5] Lifshits M. A. and Linde W. 2002 Mem. Amer. Math. Soc.157 (Providence, RI: Amer. Math. Soc.) · Zbl 0999.47034 [6] Oinarov R. 1993 On weighted norm inequalities with three weights J. London Math. Soc. (2)48 103-116 · Zbl 0811.26008 [7] Stepanov V. D. and Ushakova E. P. 2001 On integral operators with variable limits of integration Proc. Steklov Inst. Math.232 298-317 [8] Lizorkin P. I. and Otelbaev M. 1979 Imbedding theorems and compactness for spaces of Sobolev type with weights Mat. Sb.108(150) 358-377 · Zbl 0432.46029 [9] Lizorkin P. I. and Otelbaev M. 1980 Imbedding theorems and compactness for spaces of Sobolev type with weights. II Mat. Sb.112(154) 56-85 · Zbl 0465.46031 [10] Lizorkin P. I. and Otelbaev M. O. 1984 Estimates of approximate numbers of the imbedding operators for spaces of Sobolev type with weights Proc. Steklov Inst. Math.170 213-232 · Zbl 0572.46037 [11] Triebel H. 1978 North-Holland Math. Library18 (Berlin: VEB Deutscher Verlag der Wissenschaften) [12] Triebel H. 1975 Interpolation properties of \(\varepsilon \)-entropy and diameters. Geometric characteristics of imbedding for function spaces of Sobolev-Besov type Mat. Sb,98(140) 27-41 · Zbl 0351.46024 [13] Bojkov I. V. 1998 Approximation of some classes of functions by local splines Zh. Vychisl. Mat. Mat. Fiz.38 25-33 · Zbl 0954.41017 [14] Mynbaev K. T. and Otelbaev M. O. 1988 Weighted function spaces and the spectrum of differential operators (Moscow: Nauka) · Zbl 0651.46037 [15] Aitenova M. S. and Kusainova L. K. 2002 On the asymptotics of the distribution of approximation numbers of embeddings of weighted Sobolev classes. I Matem. Zhurn.2 3-9 · Zbl 1134.46307 [16] Aitenova M. S. and Kusainova L. K. 2002 On the asymptotics of the distribution of approximation numbers of embeddings of weighted Sobolev classes. II Matem. Zhurn.2 7-14 · Zbl 1134.46308 [17] Vasil’eva A. A. 2020 Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain with conditions on the derivatives of order \(r\) and zero arXiv:2004.06013v2 [18] Tikhomirov V. M. 1976 Some problems in approximation theory (Moscow: Moscow State University) [19] Tikhomirov V. M. 1987 Approximation theory Analiz - 214 103-260 · Zbl 0655.41002 [20] Pinkus A. 1985 Ergeb. Math. Grenzgeb. (3)7 (Berlin: Springer-Verlag) [21] Pietsch A. \(1974 s\)-numbers of operators in Banach spaces Studia Math.51 201-223 · Zbl 0294.47018 [22] Stesin M. I. 1975 Aleksandrov diameters of finite-dimensional sets and classes of smooth functions Dokl. Akad. Nauk SSSR220 1278-1281 [23] Ismagilov R. S. 1974 Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials Uspekhi Mat. Nauk29 161-178 · Zbl 0303.41039 [24] Gluskin E. D. 1983 Norms of random matrices and widths of finite-dimensional sets Mat. Sb.120(162) 180-189 · Zbl 0558.46013 [25] Kašin B. S. 1977 Diameters of some finite-dimensional sets and classes of smooth functions Izv. Akad. Nauk SSSR Ser. Mat.41 334-351 · Zbl 0378.46027 [26] Maiorov V. E. 1975 Discretization of the problem of diameters Uspekhi Mat. Nauk30 179-180 [27] Garnaev A. Yu. and Gluskin E. D. 1984 On widths of the Euclidean ball Dokl. Akad. Nauk SSSR277 1048-1052 [28] Kashin B. S. 1975 The diameters of octahedra Uspekhi Mat. Nauk30 251-252 [29] Galeev È. M. 1981 The Kolmogorov diameter of the intersection of classes of periodic functions and of finite-dimensional sets Mat. Zametki29 749-760 · Zbl 0507.41020 [30] Galeev È. M. 1995 Kolmogorov \(n\)-width of some finite-dimensional sets in a mixed measure Mat. Zametki58 144-148 · Zbl 0865.41022 [31] Galeev È. M. 1990 Kolmogorov widths of classes of periodic functions of one and several variables Izv. Akad. Nauk SSSR Ser. Mat.54 418-430 · Zbl 0728.42002 [32] Galeev È. M. 2011 Widths of function classes and finite-dimensional sets Vladikavkaz. Mat. Zh.13 3-14 · Zbl 1238.46025 [33] Gluskin E. D. 1987 Intersections of a cube with an octahedron are poorly approximated by low-dimensional subspaces Approximation of Functions by Special Classes of Operators 35-41 · Zbl 0685.46007 [34] Izaak A. D. 1994 Kolmogorov widths in finite-dimensional spaces with mixed norms Mat. Zametki55 43-52 · Zbl 0879.46004 [35] Izaak A. D. 1996 Widths of Hölder-Nikol’skii classes and finite-dimensional subsets in spaces with mixed norm Mat. Zametki59 459-461 · Zbl 0929.46025 [36] Malykhin Yu. V. and Ryutin K. S. 2017 The product of octahedra is badly approximated in the \(\ell_{2,1}\)-metric Mat. Zametki101 85-90 · Zbl 1371.52008 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.