Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-28T01:32:42.849Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak weight-semi-greedy Markushevich bases

Part of: Approximations and expansions Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 June 2023

Miguel Berasategui Open the ORCID record for Miguel Berasategui [Opens in a new window]  and
Silvia Lassalle Open the ORCID record for Silvia Lassalle [Opens in a new window]
Miguel Berasategui
Affiliation:
IMAS–UBA–CONICET - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (mberasategui@dm.uba.ar)
Silvia Lassalle
Affiliation:
Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284, 1644 Victoria, Buenos Aires, Argentina IMAS–CONICET, Intendente Güiraldes S/N, 1428, Buenos Aires, Argentina (slassalle@udesa.edu.ar)
Get access Rights & Permissions [Opens in a new window]

Abstract

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

Keywords

basesgreedy approximationweights

MSC classification

Primary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Secondary: 46B15: Summability and bases 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 62
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albiac, F. and Ansorena, J. L.. Characterization of 1-quasi-greedy bases. J. Approx. Theory 201 (2016), 7–12.CrossRefGoogle Scholar
Albiac, F. and Ansorena, J. L.. Characterization of 1-almost-greedy bases. Rev. Mat. Complutense 30 (2017), 13–24.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L. and Berná., P. M. Forum of Mathematics, Sigma 10 (2022), E113.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L., Berasategui, M., Berná, P. M. and Lassalle, S.. Bidemocratic Bases and Their Connections with Other Greedy-Type Bases, Constr. Approx. 57 (2023), 125–160.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L., Berasategui, M., Berná, P. M. and Lassalle, S., Weak forms of unconditionality of bases in greedy approximation, Stud. Math. 267 (2022), 1–17.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L., Berná, P. M. and Wojtaszczyk, P.. Greedy approximation for biorthogonal systems in quasi-Banach spaces. Dissertationes Math. 560 (2021), 1–88.Google Scholar
Albiac, F., Ansorena, J. L., Garrigós, G., Hernández, H., Raja, M.. Conditionality constants of quasi-greedy bases in super-reflexive Banach spaces. Stud. Math. 227 (2015), 133–140.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L., Dilworth, S. and Kutzarova, D.. Building highly conditional almost greedy and quasi-greedy bases in Banach spaces. J. Funct. Anal. 276 (2019), 1893–1924.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L. and Wojtaszczyk, P.. Conditional quasi-greedy bases in non-superreflexive banach spaces. Constr. Approx. 49 (2019), 103–122.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L. and Wojtaszczyk, P.. Quasi-greedy bases in $\ell _p$ ($0< p<1$) are democratic. J. Funct. Anal. 280 (2020), 108871.CrossRefGoogle Scholar
Albiac, F. and Kalton, N. J.. Topics in Banach space theory. Second, Graduate Texts in Mathematics, Vol. 233 (Springer, Cham, 2016). With a foreword by Gilles Godefory.CrossRefGoogle Scholar
Albiac, F. and Wojtaszscyk, P.. Characterization of 1-greedy bases. J. Approx. Theory 201 (2006), 65–86.CrossRefGoogle Scholar
Berasategui, M. and Berná, P. M.. Quasi-greedy bases for sequences with gaps. Nonlinear Anal. Theory Methods Appl. 208 (2021), 112294.CrossRefGoogle Scholar
Berasategui, M. and Lassalle, S.. Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases. J. Fourier Anal. Appl. 29 (2023), 20.CrossRefGoogle Scholar
Berná, P.. Equivalence between almost greedy and semi-greedy bases. J. Math. Anal. Appl. 470 (2019), 218–225.CrossRefGoogle Scholar
Berná, P.. Characterization of weight-semi-greedy bases. J. Fourier Anal. Appl. 26 (2020), 1–21.CrossRefGoogle Scholar
Berná, P. and Blasco, Ó.. Characterization of greedy bases in Banach spaces. J. Approx. Theory 215 (2017), 28–39.CrossRefGoogle Scholar
Berná, P. M., Blasco, O. and Garrigós, G.. Lebesgue inequalities for greedy algorithm in general bases. Rev. Mat. Complutense 30 (2017), 369–392.CrossRefGoogle Scholar
Berná, P. M., Blasco, O., Garrigós, G., Hernández, E. and Oikhberg, T.. Embeddings and Lebesgue-type inequalities for the greedy algorithm in Banach spaces. Constr. Approx. 48 (2018), 415–451.CrossRefGoogle Scholar
Berná, P., Dilworth, S. J., Kutzarova, D., Oikhberg, T. and Wallis, B.. The weighted property (A) and the greedy algorithm. J. Approx. Theory 248 (2019), 105300.CrossRefGoogle Scholar
Dilworth, S. J., Garrigós, G., Hernández, E., Kutzarova, D. and Temlyakov, V.. Lebesgue-type inequalities in greedy approximation. J. Funct. Anal. 280 (2021), 108885.CrossRefGoogle Scholar
Dilworth, S. J., Hoffmann, M. and Kutzarova, D.. Non-equivalent greedy and almost greedy bases in $\ell _p$. J. Funct. Spaces Appl. 4 (2006), 25–42.CrossRefGoogle Scholar
Dilworth, S. J., Kalton, N. J. and Kutzarova, D.. On the existence of almost greedy bases in Banach spaces. Stud. Math. 159 (2003), 67–101.CrossRefGoogle Scholar
Dilworth, S. J., Kalton, N. J., Kutzarova, D. and Temlyakov, V. N.. The thresholding greedy algorithm, greedy bases, and duality. Constr. Approx. 19 (2003), 575–597.CrossRefGoogle Scholar
Dilworth, S. J., Kutzarova, D. and Oikhberg, T.. Lebesgue constants for the weak greedy algorithm. Rev. Mat. Complutense 28 (2015), 393–409.CrossRefGoogle Scholar
Dilworth, S. J., Kutzarova, D., Schlumprecht, T. and Wojtaszczyk, P.. Weak thresholding greedy algorithms in Banach spaces. J. Funct. Anal. 263 (2012), 3900–3921.CrossRefGoogle Scholar
Dilworth, S. J., Kutzarova, D., Temlyakov, V. N. and Wallis, B.. Weight-almost greedy bases. Proc. Steklov Inst. Math. 303 (2018), 109–128.CrossRefGoogle Scholar
Garrigós, G., Hernández, E. and Oikhberg, T.. Lebesgue type inequalities for quasi-greedy bases. Constr. Approx. 38 (2013), 447–470.CrossRefGoogle Scholar
Gogyan, S.. On convergence of weak thresholding greedy algorithm in $L_1(0,1)$. J. Approx. Theory 161 (2009), 49–64.CrossRefGoogle Scholar
Kerkyacharian, G., Picard, D. and Temlyakov, V. N.. Some inequalities for the tensor product of greedy bases and weight-greedy bases. East J. Approx. 12 (2006), 103–118.Google Scholar
Konyagin, S. V. and Temlyakov, V. N.. A remark on greedy approximation in Banach spaces. East J. Approx. 3 (1999), 365–379.Google Scholar
Konyagin, S. V. and Temlyakov, V. N.. Greedy approximation with regard to bases and general minimal systems. Serdica Math. J. 28 (2002), 305–328.Google Scholar
Konyagin, S. V. and Temlyakov, V. N.. Convergence of greedy approximation. I. General systems. Stud. Math. 159 (2003), 143–160.CrossRefGoogle Scholar
Oikhberg, T.. Greedy algorithms with gaps. J. Approx. Theory 255 (2018), 176–190.CrossRefGoogle Scholar
Oswald, P.. Greedy algorithms and best $m$-term approximation with respect to biorthogonal systems. J. Fourier Anal. Appl. 7 (2001), 325–341.CrossRefGoogle Scholar
Temlyakov, V. N.. Greedy algorithm and $m$-term trigonometric approximation. Constr. Approx. 14 (1998), 569–587.CrossRefGoogle Scholar
Temlyakov, V. N.. Greedy approximation. Acta Numer. 17 (2008), 235–409.CrossRefGoogle Scholar
Temlyakov, V. N.. Greedy approximation, Vol. 20 (Cambridge University Press, Cambridge 2011).CrossRefGoogle Scholar
Temlyakov, V. N.. Constructive sparse trigonometric approximations and other problems for functions with mixed smoothness. Sb. Math. 203 (2015), 1628–1656.CrossRefGoogle Scholar
Wojtaszczyk, P.. Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107 (2000), 293–314.CrossRefGoogle Scholar