Geodesic
On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 20-28.

Voir la notice de l'article provenant de la source Math-Net.Ru

Based on the study of the geometric properties of unconditional quasibasic sequences, we show that in an arbitrary symmetric space there exists no unconditional quasibasis consisting of nonnegative functions. Moreover, we demonstrate that in an arbitrary Banach function lattice $X$ of type $p>1$ one can introduce an equivalent norm such that there exists no monotone (with respect to the new norm) basis in $X$ that consists of nonnegative functions.
Keywords: basis, quasibasis, basic sequence, symmetric space, Rademacher system, type of a Banach space. 
@article{TM_2022_319_a1,
     author = {S. V. Astashkin and P. A. Terekhin},
     title = {On {Quasibases} and {Bases} of {Symmetric} {Spaces} {Consisting} of {Nonnegative} {Functions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {20--28},
     publisher = {mathdoc},
     volume = {319},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_319_a1/}
}
TY  - JOUR
AU  - S. V. Astashkin
AU  - P. A. Terekhin
TI  - On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2022
SP  - 20
EP  - 28
VL  - 319
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2022_319_a1/
LA  - ru
ID  - TM_2022_319_a1
ER  - 
%0 Journal Article
%A S. V. Astashkin
%A P. A. Terekhin
%T On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2022
%P 20-28
%V 319
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2022_319_a1/
%G ru
%F TM_2022_319_a1
S. V. Astashkin; P. A. Terekhin. On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 20-28. http://geodesic.mathdoc.fr/item/TM_2022_319_a1/

[1] Albiac F., Kalton N.J., Topics in Banach space theory, Grad. Texts Math., 233, Springer, New York, 2006 | MR

[2] Astashkin S.V., “$\Lambda (p)$-spaces”, J. Funct. Anal., 266:8 (2014), 5174–5198 | DOI | MR

[3] S. V. Astashkin, The Rademacher System in Function Spaces, Birkhäuser, Cham, 2020 | MR

[4] Astashkin S.V., Terekhin P.A., “Representing systems of dilations and translations in symmetric function spaces”, J. Fourier Anal. Appl., 26:1 (2020), 13 | DOI | MR

[5] Carothers N.L., A short course on Banach space theory, LMS Stud. Texts, 64, Cambridge Univ. Press, Cambridge, 2005 | MR

[6] J. Diestel, Geometry of Banach Spaces: Selected Topics, Lect. Notes Math., 485, Springer, Berlin, 1975 | DOI | MR

[7] Faber G., “Über die Orthogonalfunktionen des Herrn Haar”, Deutsche Math.-Ver., 19 (1910), 104–112

[8] Filippov V.I., Oswald P., “Representation in $L_p$ by series of translates and dilates of one function”, J. Approx. Theory, 82:1 (1995), 15–29 | DOI | MR

[9] Johnson W.B., Schechtman G., “A Schauder basis for $L_1(0,\infty )$ consisting of non-negative functions”, Ill. J. Math., 59:2 (2015), 337–344 | MR

[10] Kadec M.I., Pełczyński A., “Bases, lacunary sequences and complemented subspaces in the spaces $L_p$”, Stud. math., 21 (1962), 161–176 | DOI | MR

[11] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982 | MR | MR

[12] B. S. Kashin and A. A. Saakyan, Orthogonal Series, Transl. Math. Monogr., 75, Am. Math. Soc., Providence, RI, 1989 | MR

[13] Kazarian K.S., Zink R.E., “Subsystems of the Schauder system that are quasibases for $L^p[0,1]$, $1\le p+\infty $”, Proc. Amer. Math. Soc., 126:10 (1998), 2883–2893 | DOI | MR

[14] S. G. Kreĭn, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr., 54, Am. Math. Soc, Providence, RI, 1982 | MR

[15] Lindenstrauss J., Tzafriri L., Classical Banach spaces. II: Function spaces, Springer, Berlin, 1979 | MR

[16] Powell A.M., Spaeth A.H., “Nonnegativity constraints for structured complete systems”, Trans. Amer. Math. Soc., 368:8 (2016), 5783–5806 | DOI | MR

[17] A. A. Talalyan, “The representation of measurable functions by series”, Russ. Math. Surv., 15:5 (1960), 75–136 | DOI | MR

[18] E. V. Tokarev, “Subspaces of symmetric spaces of functions”, Funct. Anal. Appl., 13:2 (1979), 152–153 | DOI | MR

[19] P. L. Ul'yanov, “On series with respect to a Schauder system”, Math. Notes, 7:4 (1970), 261–268 | DOI