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

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

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

Bibliographic databases:

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

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
2. D. B. Bazarkhanov, “Nonlinear trigonometric approximations of multivariate function classes”, Proc. Steklov Inst. Math., 293 (2016), 2–36
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
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
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
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
7. S. A. Stasyuk, “Razrezhennoe trigonometricheskoe priblizhenie klassov Besova funktsii s maloi smeshannoi gladkostyu”, Tr. IMM UrO RAN, 23, no. 3, 2017, 244–252
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
9. V. Temlyakov, “Constructive sparse trigonometric approximation for functions with small mixed smoothness”, Constr. Approx., 45:3 (2017), 467–495
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
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
12. V. Temlyakov, “On the entropy numbers of the mixed smoothness function classes”, J. Approx. Theory, 217 (2017), 26–56
13. V. N. Temlyakov, “The Marcinkiewicz-type discretization theorems”, Constr. Approx., 48:2 (2018), 337–369
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