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Estimates for the widths of discrete function classes generated by a two-weight summation operator

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Abstract

Order estimates are obtained for the Kolmogorov, linear, and Gel’fand widths of the image of the unit ball of the space l p under the action of a two-weight summation operator with weights of special kind. Some limit conditions on the parameters defining the weights are considered.

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References

  1. G. Bennett, “Some elementary inequalities. III, ” Q. J. Math. Oxford, Ser.2, 42 (166), 149–174 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  2. 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).

    Article  MathSciNet  MATH  Google Scholar 

  3. W. D. Evans, D. J. Harris, and J. Lang, “The approximation numbers of Hardy-type operators on trees, ” Proc. London Math. Soc., Ser. 3, 83 (2), 390–418 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Yu. Garnaev and E. D. Gluskin, “On widths of the Euclidean ball, ” Dokl. Akad. Nauk SSSR 277 (5), 1048–1052 (1984) [Sov. Math., Dokl. 30, 200–204 (1984)].

    MathSciNet  MATH  Google Scholar 

  5. E. D. Gluskin, “Norms of random matrices and widths of finite-dimensional sets, ” Mat. Sb. 120 (2), 180–189 (1983) [Math. USSR, Sb. 48, 173–182 (1984)].

    MathSciNet  MATH  Google Scholar 

  6. 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).

    Article  MathSciNet  MATH  Google Scholar 

  7. D. D. Haroske and L. Skrzypczak, “Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights. II: General weights, ” Ann. Acad. Sci. Fenn., Math. 36 (1), 111–138 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  8. B. S. Kashin, “On Kolmogorov diameters of octahedra, ” Dokl. Akad. Nauk SSSR 214 (5), 1024–1026 (1974) [Sov. Math., Dokl. 15, 304–307 (1974)].

    MathSciNet  Google Scholar 

  9. B. S. Kashin, “The diameters of octahedra, ” Usp. Mat. Nauk 30 (4), 251–252 (1975).

    MathSciNet  Google Scholar 

  10. B. S. Kashin, “Diameters of some finite-dimensional sets and classes of smooth functions, ” Izv. Akad. Nauk SSSR, Ser. Mat. 41 (2), 334–351 (1977) [Math. USSR, Izv. 11, 317–333 (1977)].

    MathSciNet  Google Scholar 

  11. M. A. Lifshits, “Bounds for entropy numbers for some critical operators, ” Trans. Am. Math. Soc. 364 (4), 1797–1813 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. A. Lifshits and W. Linde, Approximationand Entropy Numbers of Volterra Operators with Application to Brownian Motion (Am. Math. Soc., Providence, RI, 2002), Mem. AMS 157 (745).

    MATH  Google Scholar 

  13. M. Lifshits and W. Linde, “Compactness properties of weighted summation operators on trees, ” Stud. Math. 202 (1), 17–47 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Lifshits and W. Linde, “Compactness properties of weighted summation operators on trees—the critical case, ” Stud. Math. 206 (1), 75–96 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  15. E. N. Lomakina and V. D. Stepanov, “On asymptotic behaviour of the approximation numbers and estimates of Schatten–von Neumann norms of the Hardy-type integral operators, ” in Function Spaces and Applications: Proc. Int. Conf., New Delhi, 1997 (Narosa Publ. House, New Delhi, 2000), pp. 153–187.

    Google Scholar 

  16. E. N. Lomakina and V. D. Stepanov, “Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann–Liouville operator, ” Mat. Tr. 9 (1), 52–100 (2006) [Sib. Adv. Math. 17 (1), 1–36 (2007)].

    MathSciNet  MATH  Google Scholar 

  17. A. Pietsch, “s-Numbers of operators in Banach spaces, ” Stud. Math. 51, 201–223 (1974).

    MathSciNet  MATH  Google Scholar 

  18. A. I. Stepanets, Methodsof Approximation Theory (Koninklijke Brill NV, Leiden, 2005).

    Book  Google Scholar 

  19. M. I. Stesin, “Aleksandrov diameters of finite-dimensional sets and classes of smooth functions, ” Dokl. Akad. Nauk SSSR 220 (6), 1278–1281 (1975) [Sov. Math., Dokl. 16, 252–256 (1975)].

    MathSciNet  MATH  Google Scholar 

  20. V. M. Tikhomirov, “Diameters of sets in function spaces and the theory of best approximations, ” Usp. Mat. Nauk 15 (3), 81–120 (1960) [Russ. Math. Surv. 15 (3), 75–111 (1960)].

    Google Scholar 

  21. A. A. Vasil’eva, “Widths of weighted Sobolev classes on a John domain: Strong singularity at a point, ” Rev. Mat. Complut. 27 (1), 167–212 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. A. Vasil’eva, “Widths of function classes on sets with tree-like structure, ” J. Approx. Theory 192, 19–59 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. A. Vasil’eva, “Widths of Sobolev weight classes on a domain with cusp, ” Mat. Sb. 206 (10), 37–70 (2015) [Sb. Math. 206, 1375–1409 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  24. A. A. Vasil’eva, “Estimates for n-widths of two-weighted summation operators on trees, ” Math. Notes 99, 243–252 (2016).

    Article  MathSciNet  MATH  Google Scholar 

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. A. Vasil’eva

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  1. A. A. Vasil’eva
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Original Russian Text © A.A. Vasil’eva, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 294, pp. 308–324.

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Vasil’eva, A.A. Estimates for the widths of discrete function classes generated by a two-weight summation operator. Proc. Steklov Inst. Math. 294, 291–307 (2016). https://doi.org/10.1134/S0081543816060201

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