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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 6, Pages 21–48 (Mi izv8220)  

This article is cited in 8 scientific papers (total in 8 papers)

Density of a semigroup in a Banach space

P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study conditions on a set $M$ in a Banach space $X$ which are necessary or sufficient for the set $R(M)$ of all sums $x_1+…+x_n$, $x_k\in M$, to be dense in $X$. We distinguish conditions under which the closure $\overline{R(M)}$ is an additive subgroup of $X$, and conditions under which this additive subgroup is dense in $X$. In particular, we prove that if $M$ is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space $X$, and does not lie in a closed half-space $\{x\in X\colon f(x)\ge0\}$, $f\in X^*$, and is minimal in the sense that every proper subarc of $M$ lies in an open half-space $\{x\in X\colon f(x)>0\}$, then $\overline{R(M)}=X$. We apply our results to questions of approximation in various function spaces.

Keywords: Banach space, additive semigroup, density, uniformly convex space, modulus of smoothness, approximation, simple partial fractions.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation НШ-3682.2014.1
Russian Foundation for Basic Research 14-01-00510
14-01-91158


DOI: https://doi.org/10.4213/im8220

Full text: PDF file (653 kB)
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English version:
Izvestiya: Mathematics, 2014, 78:6, 1079–1104

Bibliographic databases:

UDC: 517.982.256+517.538.5
MSC: 41A65, 46B20, 46B25
Received: 03.02.2014
Revised: 21.04.2014

Citation: P. A. Borodin, “Density of a semigroup in a Banach space”, Izv. RAN. Ser. Mat., 78:6 (2014), 21–48; Izv. Math., 78:6 (2014), 1079–1104

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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. P. A. Borodin, O. N. Kosukhin, “Quantitative Expressions for the Connectedness of Sets in ${\mathbb R}^n$”, Math. Notes, 98:5 (2015), 707–713  mathnet  crossref  crossref  mathscinet  isi  elib
    2. P. A. Borodin, “Approximation by simple partial fractions with constraints on the poles. II”, Sb. Math., 207:3 (2016), 331–341  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. S. M. Tabatabaie, “The problem of density on $L^{2}(G)$”, Acta Math. Hungar., 150:2 (2016), 339–345  crossref  mathscinet  zmath  isi  elib  scopus
    4. P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. Math., 81:6 (2017), 1080–1094  mathnet  crossref  crossref  adsnasa  isi  elib
    5. P. A. Borodin, S. V. Konyagin, “Convergence to zero of exponential sums with positive integer coefficients and approximation by sums of shifts of a single function on the line”, Anal. Math., 44:2 (2018), 163–183  crossref  mathscinet  zmath  isi  scopus
    6. P. A. Borodin, “Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk”, Math. Notes, 104:1 (2018), 3–9  mathnet  crossref  crossref  isi  elib
    7. P. A. Borodin, “Density of sums of shifts of a single vector in sequence spaces”, Proc. Steklov Inst. Math., 303 (2018), 31–35  mathnet  crossref  crossref  isi  elib
    8. S. M. Tabatabaie, “The problem of density on commutative strong hypergroups”, Math. Rep., 20:3 (2018), 227–232  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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