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Mat. Sb., 2015, Volume 206, Number 11, Pages 131–160 (Mi msb8466)  

This article is cited in 13 scientific papers (total in 13 papers)

Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness

V. N. Temlyakovab

a University of South Carolina, Columbia, SC, USA
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use a technique based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s–1990s, and recent results on greedy approximation to obtain sharp estimates for best $m$-term approximation with respect to the trigonometric system. We give some observations on the numerical integration and approximate recovery of functions with mixed smoothness. We prove lower bounds, which show that one cannot improve the accuracy of sparse grids methods with $\asymp 2^nn^{d-1}$ points in the grid by adding $2^n$ arbitrary points. In the case of numerical integration these lower bounds provide the best available lower bounds for optimal cubature formulae and for sparse grids based cubature formulae.
Bibliography: 31 titles.

Keywords: nonlinear approximation, sparse approximation, trigonometric system, constructive methods.

Funding Agency Grant Number
National Science Foundation DMS-1160841
This research was supported by the NSF (grant no. DMS-1160841).


DOI: https://doi.org/10.4213/sm8466

Full text: PDF file (679 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2015, 206:11, 1628–1656

Bibliographic databases:

UDC: 517.518.8
MSC: Primary 41A60, 42A10, 46E35; Secondary 41A65
Received: 31.12.2014

Citation: V. N. Temlyakov, “Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness”, Mat. Sb., 206:11 (2015), 131–160; Sb. Math., 206:11 (2015), 1628–1656

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. A. Stasyuk, “Priblizhenie nekotorykh gladkostnykh klassov periodicheskikh funktsii mnogikh peremennykh polinomami po tenzornoi sisteme Khaara”, Tr. IMM UrO RAN, 21, no. 4, 2015, 251–260  mathnet  mathscinet  elib
    2. D. B. Bazarkhanov, “Nonlinear trigonometric approximations of multivariate function classes”, Proc. Steklov Inst. Math., 293 (2016), 2–36  mathnet  crossref  crossref  mathscinet  isi  elib
    3. S. A. Stasyuk, “Konstruktivnye razrezhennye trigonometricheskie priblizheniya dlya klassov funktsii s nebolshoi smeshannoi gladkostyu”, Tr. IMM UrO RAN, 22, no. 4, 2016, 247–253  mathnet  crossref  mathscinet  elib
    4. S. A. Stasyuk, “Best $m$-term trigonometric approximation for periodic functions with low mixed smoothness from the Nikol'skii-Besov-type classes”, Ukr. Math. J., 68:7 (2016), 1121–1145  crossref  mathscinet  isi  scopus
    5. M. Ullrich, T. Ullrich, “The role of Frolov's cubature formula for functions with bounded mixed derivative”, SIAM J. Numerical Anal., 2016, no. 2, 969–993  crossref  mathscinet  zmath  isi  scopus
    6. V. Temlyakov, “Sparse approximation by greedy algorithms”, Mathematical analysis, probability and applications—plenary lectures, Springer Proc. Math. Stat., 177, Springer, Cham, 2016, 183–215  crossref  mathscinet  zmath  isi  scopus
    7. S. A. Stasyuk, “Razrezhennoe trigonometricheskoe priblizhenie klassov Besova funktsii s maloi smeshannoi gladkostyu”, Tr. IMM UrO RAN, 23, no. 3, 2017, 244–252  mathnet  crossref  elib
    8. V. K. Nguyen, M. Ullrich, T. Ullrich, “Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube”, Constr. Approx., 46:1 (2017), 69–108  crossref  mathscinet  zmath  isi  scopus
    9. V. Temlyakov, “Constructive sparse trigonometric approximation for functions with small mixed smoothness”, Constr. Approx., 45:3 (2017), 467–495  crossref  mathscinet  zmath  isi  scopus
    10. D. B. Bazarkhanov, “Sparse approximation of some function classes with respect to multiple Haar system on the unit cube”, International Conference Functional Analysis In Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 030017  crossref  isi  scopus
    11. G. Akishev, “Estimations of the best $M$-term approximations of functions in the Lorentz space with constructive methods”, Vestnik Karagandinskogo un-ta. Ser. Matem., 2017, no. 3(87), 13–26 http://mathematics-vestnik.ksu.kz/ru/content/srch/2017_Mathematics_3_86_2017.pdf  isi
    12. V. Temlyakov, “On the entropy numbers of the mixed smoothness function classes”, J. Approx. Theory, 217 (2017), 26–56  crossref  mathscinet  zmath  isi  scopus
    13. V. N. Temlyakov, “The Marcinkiewicz-type discretization theorems”, Constr. Approx., 48:2 (2018), 337–369  crossref  mathscinet  isi  scopus
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