Abstract
Our earlier one-dimensional results concerning Kolmogorov widths of weighted Sobolev classes are extended to the multidimensional case.
Similar content being viewed by others
References
A. Kufner, Weighted Sobolev Spaces, Teubner-Texte Math., 31 (BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978; Mir, Moscow, 1980).
B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces (Lecture Notes Math, 1736, Springer-Verlag, Berlin, 2000).
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators Cambridge Tracts in Mathematics 120 (Cambridge University Press, Cambridge, 1996).
H. Triebel, Theory of Function Spaces III (Birkhäuser Verlag, 2006).
D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings (Springer, 2004).
L. D. Kudryavtsev and S. M. Nikol’skii, “Spaces of Differentiable Functions of Several Variables and Embedding Theorems,” Current problems in mathematics. Fundamental directions 26, 5–157 (1988) [Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, in Russian].
V. D. Stepanov, “Two-Weighted Estimates for Riemann-Liouville Integrals,” Rept. 39 (Českoslov. Akad. Věd. (Mat. Ústav.) Praha, 1998), pp. 1–28.
E. N. Batuev and V. D. Stepanov, “Weighted Inequalities of Hardy Type,” Sibirsk. Mat. Zh. 30(1), 13–22 (1989) [Siberian Math. J. 30 (1), 8–16 (1989)].
V. D. Stepanov, “Two-Weighted Estimates for Riemann-Liouville Integrals,” Izv. Akad. Nauk SSSR Ser. Mat. 54(3), 645–656 (1990) [Math. USSR-Izv. 36 (3), 669–681 (1991)].
D. V. Prokhorov, “On the Boundedness and Compactness of a Class of Integral Operators,” J. London Math. Soc. 61, 617–628 (2000).
K. F. Andersen and H. P. Heinig, “Weighted Norm Inequalities for Certain Integral Operators,” SIAM J. Math. Anal. 14(4), 834–844 (1983).
H. P. Heinig, “Weighted Norm Inequalities for Certain Integral Operators. II,” Proc. Amer. Math. Soc. 95(3), 387–395 (1985).
K. F. Andersen and E. T. Sawyer, “Weighted Norm Inequalities for the Riemann-Liouville and Weyl Fractional Integral Operators,” Trans. Amer. Math. Soc. 308(2), 547–558 (1988).
D. V. Prokhorov and V. D. Stepanov, “Weighted Estimates for Riemann-Liouville Operators and Their Applications,” Tr. Mat. Inst. Steklova 243, 289–312 (2003) [Proc. Steklov Inst. Math. 4 (243), 278–301 (2004)].
L. D. Kudryavtsev,“Direct and Inverse Imbedding Theorems. Applications to the Solution of Elliptic Equations by Variational Methods,” Tr. Mat. Inst. Steklova 55, 182 (1959) [Russian].
P. I. Lizorkin and M. Otelbaev, “Imbedding and Compactness Theorems for Sobolev-Type Spaces with Weights. I, II,” Mat. Sb. 108(1), 358–377 (1979); 112 (1), 56–85 (1980) [Math. USSR-Sb. 40 (1), 51–77 (1981)].
P. Gurka and B. Opic, “Continuous and Compact Embeddings of Weighted Sobolev Spaces. I, II, III,” Czech. Math. J. 38(113), 730–744 (1988); 39 (114), 78–94 (1989); 41 (116), 317–341 (1991).
O. V. Besov, “On the Compactness of Embeddings of Weighted Sobolev Spaces on a Domain with an Irregular Boundary,” Tr. Mat. Inst. Steklova 232, 72–93 (2001) [Proc. Steklov Inst. Math. 1 (232), 66–87 (2001)].
O. V. Besov, “Sobolev’s Embedding Theorem for a Domain with an Irregular Boundary,” Mat. Sb. 192(3), 3–26 (2001) [Sb. Math. 192 (3–4), 323–346 (2001)].
O. V. Besov, “On the Compactness of Embeddings of Weighted Sobolev Spaces on a Domain with an Irregular Boundary,” Dokl. Akad. Nauk 376(6), 727–732 (2001) (Russian).
O. V. Besov, “Integral Estimates for Differentiable Functions on Irregular Domains,” Mat. Sb. 201(12), 69–82 (2010) [Sb. Math. 201 (12), 1777–1790 (2010)].
F. Antoci, “Some Necessary and Some Sufficient Conditions for the Compactness of the Embedding of Weighted Sobolev Spaces,” Ricerche Mat. 52(1), 55–71 (2003).
V. Gol’dshtein and A. Ukhlov, “Weighted Sobolev Spaces and Embedding Theorems,” Trans. Amer. Math. Soc. 361(7), 3829–3850 (2009).
S. Heinrich, “On the Relation between Linear n-Widths and Approximation Numbers,” J. Approx. Theory 58(3), 315–333 (1989).
V. M. Tihomirov [Tikhomirov], “Diameters of Sets in Functional Spaces and the Theory of Best Approximations,” Uspekhi Mat. Nauk 15(3), 81–120 (1960) [Russian Math. Surveys 15 (3), 75–111 (1960)].
V. M. Tihomirov [Tikhomirov] and S. B. Babadzanov, “Diameters of a Function Class in an L p-Space (p ≥ 1),” Izv. Akad. Nauk UzSSR Ser. Fiz. Mat. Nauk 11(2), 24–30 (1967) (Russian).
A. P. Buslaev and V. M. Tikhomirov, “The Spectra of Nonlinear Differential Equations and Widths of Sobolev Classes,” Mat. Sb. 181(12), 1587–1606 (1990) [Math. USSR-Sb. 71 (2), 427–446 (1992)].
R. S. Ismagilov, “Diameters of Sets in Normed Linear Spaces, and the Approximation of Functions by Trigonometric Polynomials,” Uspekhi Mat. Nauk 29(3), 161–178 (1974) [Russ. Math. Surv. 29 (3), 169–186 (1974)].
B. S. Kashin [Kašin], “The Widths of Certain Finite-Dimensional Sets and Classes of Smooth Functions,” Izv. Akad. Nauk SSSR Ser. Mat. 41(2), 234–251 (1977) [Math. USSR-Izv. 11 (2), 317–333 (1977)].
V. E. Maiorov, “Discretization of the Problem of Diameters,” Uspekhi Mat. Nauk 30(6), 179–180 (1975).
Yu. I. Makovoz, “A Certain Method of Obtaining Lower Estimates for Diameters of Sets in Banach Spaces,” Mat. Sb. 87(129) (1), 136–146 (1972) [Math. USSR-Sb. 16 (1), 139–146 (1972)].
V. N. Temlyakov, “Approximation of Periodic Functions of Several Variables with Bounded Mixed Derivative,” Dokl. Akad. Nauk SSSR 253(3), 544–548 (1980) [in Russian].
V. N. Temlyakov, “Diameters of Some Classes of Functions of Several Variables,” Dokl. Akad. Nauk SSSR 267(3), 314–317 (1982) [in Russian].
V. N. Temlyakov, “Approximation of Functions with Bounded Mixed Difference by Trigonometric Polynomials, and Diameters of Certain Classes of Functions,” Izv. Akad. Nauk SSSR Ser. Mat. 46(1), 171–186 (1982) [Math. USSR-Izv. 20 (1), 173–187 (1983)].
É. M. Galeev, “Approximation of Certain Classes of Periodic Functions of Several Variables by Fourier Sums in the \(\tilde L_p \) Metric,” Uspekhi Mat. Nauk 32(4), 251–252 (1977) [in Russian].
É. M. Galeev, “The Approximation of Classes of Functions with Several Bounded Derivatives by Fourier Sums,” Mat. Zametki 23(2), 197–212 (1978) [Math. Notes 23 (2), 109–117 (1978)].
B. S. Kashin, “Widths of Sobolev Classes of Small-Order Smoothness,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5, 50–54 (1981) [Moscow Univ. Math. Bull. 36 (5), 62–66 (1981)].
E. D. Kulanin, Estimates for Diameters of Sobolev Classes of Small-Order Smoothness, Thesis. Candidate Fiz.-Math. Sciences (MGU, Moscow, 1986) [in Russian].
V. M. Tihomirov [Tikhomirov], Some Questions in Approximation Theory (Izdat. Moskov. Univ., Moscow, 1976) [in Russian].
V. M. Tikhomirov, “Approximation Theory,” in: Current problems in mathematics. Fundamental directions. vol. 14. (Itogi Nauki i Tekhniki) (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987), pp. 103–260 [in Russian].
A. Pinkus, n-Widths in Approximation Theory (Berlin: Springer, 1985).
A. Pietsch, “s-Numbers of Operators in Banach Space,” Studia Math. 51, 201–223 (1974).
M. I. Stesin, “Aleksandrov Diameters of Finite-Dimensional Sets and of Classes of Smooth Functions,” Dokl. Akad. Nauk SSSR 220(6), 1278–1281 (1975) [in Russian].
E. D. Gluskin, “Norms of Random Matrices and Diameters of Finite-Dimensional Sets,” Mat. Sb. 120(2), 180–189 (1983) [Math. USSR-Sb. 48 (1), 173–182 (1984)].
A. Yu. Garnaev and E. D. Gluskin, “The Widths of a Euclidean Ball,” Dokl. Akad. Nauk SSSR 277(5), 1048–1052 (1984) [Soviet Math. Dokl. 30 (1), 200–204 (1984)].
M. A. Lifshits and W. Linde, “Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion,” Mem. Amer. Math. Soc. 745 (2002).
D. E. Edmunds and J. Lang, “Approximation Numbers and Kolmogorov Widths of Hardy-Type Operators in a Non-Homogeneous Case,” Math. Nachr. 297(7), 727–742 (2006).
D. E. Edmunds, R. Kerman, and J. Lang, “Remainder Estimates for the Approximation Numbers of Weighted Hardy Operators Acting on L 2,” J. Anal. Math. 85, 225–243 (2001).
W. D. Evans, D. J. Harris, and J. Lang, “Two-Sided Estimates for the Approximation Numbers of Hardy-Type Operators in L ∞ and L 1,” Studia Math. 130, 171–192 (1998).
J. Lang, “Estimates for the Approximation Numbers and n-Widths of the Weighted Hardy-Type Operators,” submitted to J. Approx. Theory 2004.
J. Lang, “Improved Estimates for the Approximation Numbers of Hardy-Type Operators,” J. Approx. Theory 121(1), 61–70 (2003).
E. N. Lomakina and V. D. Stepanov, “On Asymptotic Behavior of the Approximation Numbers and Estimates of Schatten-von Neumann Norms of the Hardy-Type Integral Operators,” In: Function Spaces and Applications (Proceedings of Delhi Conference, 1997) (Narosa Publishing House, New Delhi, 2000), pp.153–187.
V. N. Konovalov and D. Leviatan, “Kolmogorov and Linear Widths of Weighted Sobolev-Type Classes on a Finite Interval,” Anal. Math. 28(4), 251–278 (2002).
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(1), 52–100 (2006) [Siberian Adv. Math. 17 (1), 1–36 (2007)].
A. A. Vasil’eva, “Estimates for the Widths of Weighted Sobolev Classes,” Mat. Sb. 201(7), 15–52 (2010) [Sb. Math. 201 (7), 947–984 (2010)].
M. Sh. Birman and M. Z. Solomyak, “Piecewise Polynomial Approximations of Functions of Classes W α p ,” Mat. Sb. 73(3), 331–355 (1967) [Math. USSR-Sb. 2 (3), 295–317 (1967)].
A. El Kolli, “n-ième épaisseur dans les espaces de Sobolev,” J. Approx. Theory 10, 268–294 (1974).
P. I. Lizorkin and M. Otelbaev, “Estimates of Approximate Numbers of the Imbedding Operators for Spaces of Sobolev Type with Weights,” Trudy Mat. Inst. Steklova 170, 213–232 (1984) [Proc. Steklov Inst. Math. 170, 245–266 (1987)].
M. O. Otelbaev, “Estimates of the Diameters in the Sense of Kolmogorov for a Class of Weighted Spaces,” Dokl. Akad. Nauk SSSR 235(6), 1270–1273 (1977) [in Russian].
M. S. Aitenova and L. K. Kusainova, “On the Asymptotics of the Distribution of Approximation Numbers of Embeddings of Weighted Sobolev Classes. I, II” [Mat. Zh. 2(1), 3–9 (2002); 2 (2), 7–14 (2002)].
A. A. Vasil’eva, “Kolmogorov Widths of Weighted Sobolev Classes on a Cube,” Trudy Inst. Mat. i Mekh. UrO RAN 16(4), 100–116 (2010).
D. D. Haroske and L. Skrzypczak, “Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I,” Rev. Mat. Complut. 21(1), 135–177 (2008).
D. D. Haroske and L. Skrzypczak, “Entropy and Approximation Numbers of Embeddings of Function SpacesWith MuckenhouptWeights, II. GeneralWeights,” Ann. Acad. Sci. Fenn. Math. 36(1), 111–138 (2011).
D. D. Haroske and L. Skrzypczak, “Entropy Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, III. Some Limiting Cases,” J. Funct. Spaces Appl. 9(2), 129–178 (2011).
A. Gąsiorowska and L. Skrzypczak, “Some s-Numbers of Embeddings of Function Spaces with Weights of Logarithmic Type” (to appear).
A. Cohen, R. DeVore, P. Petrushev, and Hong Xu, “Nonlinear Approximation and the Space BV (ℝ2),” Amer. J. Math. 121(3), 587–628 (1999).
V. G. Maz’ja [Maz’ya], Sobolev Spaces (Leningrad. Univ., Leningrad, 1985; Springer-Verlag, Berlin-New York, 1985).
G. Pisier, The Volume of Convex Bodies and Banach Space Geometry (New York: Cambridge Univ. Press, 1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by RFBR under grants nos. 09-01-00093 and 10-01-00442. Continued. The first part was published in Russian Journal of Mathematical Physics, vol. 18. no. 3, pp. 353–385 (2011). The notation is preserved and the indexing of the assertions and formulas is common.
Rights and permissions
About this article
Cite this article
Vasil’eva, A.A. Kolmogorov widths of weighted Sobolev classes on a domain for a special class of weights. II. Russ. J. Math. Phys. 18, 465–504 (2011). https://doi.org/10.1134/S1061920811040078
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920811040078