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Mat. Zametki, 1998, Volume 63, Issue 6, Pages 891–902 (Mi mz1360)  

This article is cited in 23 scientific papers (total in 23 papers)

Nonlinear Kolmogorov widths

V. N. Temlyakovab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of South Carolina

Abstract: We investigate a generalization of width in the sense of Kolmogorov suitable for estimating best $m$-term approximations. We generalize Carl's inequality which gives lower estimate of Kolmogorov widths in terms of entropy numbers. Application of these new inequalities gives some progress in the problem of estimating best $m$-term trigonometric approximations of multivariate functions.


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English version:
Mathematical Notes, 1998, 63:6, 785–795

Bibliographic databases:

UDC: 517.5
Received: 06.08.1997

Citation: V. N. Temlyakov, “Nonlinear Kolmogorov widths”, Mat. Zametki, 63:6 (1998), 891–902; Math. Notes, 63:6 (1998), 785–795

Citation in format AMSBIB
\by V.~N.~Temlyakov
\paper Nonlinear Kolmogorov widths
\jour Mat. Zametki
\yr 1998
\vol 63
\issue 6
\pages 891--902
\jour Math. Notes
\yr 1998
\vol 63
\issue 6
\pages 785--795

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    This publication is cited in the following articles:
    1. Temlyakov V.N., “Greedy algorithms with regard to multivariate systems with special structure”, Constr. Approx., 16:3 (2000), 399–425  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Dinh Dung, “Continuous algorithms in $n$-term approximation and non-linear widths”, J. Approx. Theory, 102:2 (2000), 217–242  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Andrianov A., “Nonlinear Haar Approximation of Functions with Bounded Mixed Derivative”, Wavelet Analysis and Multiresolution Methods, Proceedings, Lecture Notes in Pure and Applied Mathematics, 212, ed. He T., Marcel Dekker, 2000, 27–47  mathscinet  zmath  isi
    4. Temlyakov V.N., “Universal bases and greedy algorithms for anisotropic function classes”, Constr. Approx., 18:4 (2002), 529–550  crossref  mathscinet  zmath  isi  scopus  scopus
    5. A. S. Romanyuk, “Best $M$-term trigonometric approximations of Besov classes of periodic functions of several variables”, Izv. Math., 67:2 (2003), 265–302  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Kerkyacharian G., Picard D., “Entropy, universal coding, approximation, and bases properties”, Constr. Approx., 20:1 (2003), 1–37  crossref  mathscinet  isi  scopus  scopus
    7. DeVore R., Petrova G., Temlyakov V., “Best basis selection for approximation in $L_p$”, Found. Comput. Math., 3:2 (2003), 161–185  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Temlyakov V.N., “Nonlinear methods of approximation”, Found. Comput. Math., 3:1 (2003), 33–107  crossref  mathscinet  zmath  isi  scopus  scopus
    9. DeVore R., Kerkyacharian G., Picard D., Temlyakov V., “Approximation methods for supervised learning”, Found. Comput. Math., 6:1 (2006), 3–58  crossref  mathscinet  zmath  isi  scopus  scopus
    10. Konyagin S.V., Temlyakov V.N., “The entropy in learning theory. Error estimates”, Constr. Approx., 25:1 (2007), 1–27  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    11. Ettinger B., Sarig N., Yomdin Y., “Linear versus non-linear acquisition of step-functions”, J. Geom. Anal., 18:2 (2008), 369–399  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Temlyakov V.N., “Approximation in learning theory”, Constr. Approx., 27:1 (2008), 33–74  crossref  mathscinet  zmath  isi  scopus  scopus
    13. Proc. Steklov Inst. Math., 269 (2010), 247–258  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    14. Kurdila A.J., Xu B., “Near-Optimal Approximation Rates for Distribution Free Learning with Exponentially, Mixing Observations”, 2010 American Control Conference, Proceedings of the American Control Conference, IEEE, 2010, 504–509  crossref  isi
    15. Hansen M. Sickel W., “Best M-Term Approximation and Sobolev-Besov Spaces of Dominating Mixed Smoothness-the Case of Compact Embeddings”, Constr. Approx., 36:1 (2012), 1–51  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. Temlyakov V.N., “An Inequality for the Entropy Numbers and its Application”, J. Approx. Theory, 173 (2013), 110–121  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    17. Chkifa A., Cohen A., Schwab Ch., “Breaking the Curse of Dimensionality in Sparse Polynomial Approximation of Parametric PDEs”, J. Math. Pures Appl., 103:2 (2015), 400–428  crossref  mathscinet  zmath  isi  scopus  scopus
    18. V. N. Temlyakov, “Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness”, Sb. Math., 206:11 (2015), 1628–1656  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    19. Batenkov D., Friedland O., Yomdin Y., “Sampling, Metric Entropy, and Dimensionality Reduction”, SIAM J. Math. Anal., 47:1 (2015), 786–796  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. D. B. Bazarkhanov, “Nonlinear trigonometric approximations of multivariate function classes”, Proc. Steklov Inst. Math., 293 (2016), 2–36  mathnet  crossref  crossref  mathscinet  isi  elib
    21. Temlyakov V., “Sparse Approximation by Greedy Algorithms”, Mathematical Analysis, Probability and Applications – Plenary Lectures, Springer Proceedings in Mathematics & Statistics, Springer Proceedings in Mathematics & Statistics, 177, ed. Qian T. Rodino L., Springer, 2016, 183–215  crossref  mathscinet  zmath  isi  scopus
    22. Samorodnitsky A. Shkredov I. Yekhanin S., “Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity”, Comput. Complex., 25:2, SI (2016), 309–348  crossref  mathscinet  zmath  isi  elib  scopus
    23. Huang H., Wang H., “Greedy Algorithm With Regard to the Needlet System on the Sphere”, J. Math. Anal. Appl., 454:2 (2017), 557–570  crossref  mathscinet  zmath  isi  scopus  scopus
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