Geodesic
Quasi-greedy property of subsystems of the multivariate Haar system
Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 481-488.

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We describe all sets of dyadic cubes $\Delta=\{\Delta_k\}$ for which a subsystem of the multivariate Haar system $\{h_i\colon\operatorname{supp}({h_i})\in\Delta\}$ is quasi-greedy in $L_1(0,1)^d$. We prove that the greedy algorithm provides a good rate of convergence for those subsystems.
Keywords: greedy algorithm, quasi-greedy basis, Haar system in $L^1$, subsystem of the Haar system. 
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S. L. Gogyan. Quasi-greedy property of subsystems of the multivariate Haar system. Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 481-488. http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a1/

[1] V. Temlyakov, “Greedy approximation”, Acta Numer., 17 (2008), 235–409 | DOI | MR | Zbl

[2] S. Konyagin, V. N. Temlyakov, “A remark on greedy approximation in Banach spaces”, East J. Approx., 5:3 (1999), 365–379 | MR | Zbl

[3] P. Wojtaszczyk, “Greedy algorithm for general biorthogonal systems”, J. Approx. Theory, 107:2 (2000), 293–314 | DOI | MR | Zbl

[4] S. J. Dilworth, D. Kutzarova, P. Wojtaszczyk, “On approximate $\ell_1$ systems in Banach spaces”, J. Approx. Theory, 114:2 (2002), 214–241 | DOI | MR | Zbl

[5] S. L. Gogyan, “On greedy algorithm in $L^1(0,1)$ by regular Haar system”, J. Contemp. Math. Anal., 46:1 (2011), 21–31 | DOI | MR | Zbl

[6] S. J. Dilworth, S. Gogyan, D. Kutzarova, “On the convergence of a weak greedy algorithm for the multivariate Haar basis”, Constr. Approx., 39:2 (2014), 343–366 | DOI | MR | Zbl

[7] S. Gogyan, “Greedy algorithm with regard to Haar subsystems”, East J. Approx., 11:2 (2005), 221–236 | MR | Zbl

[8] S. J. Dilworth, N. J. Kalton, D. Kutzarova, V. N. Temlyakov, “The thresholding greedy algorithm, greedy bases, and duality”, Constr. Approx., 19:4 (2003), 575–597 | DOI | MR | Zbl