Some remarks on non-symmetric polarization
Let P:Cn→C be an m-homogeneous polynomial given by P(x)=∑1≤j1≤…≤jm≤ncj1…jmxj1…xjm. Defant and Schlüters defined a non-symmetric associated m-form LP:(Cn)m→C by LP(x(1),…,x(m))=∑1≤j1≤…≤jm≤ncj1…jmxj1 (1)…xjm (m). They estimated the norm of LP on (Cn,‖⋅‖)m by the norm of P on (Cn,‖⋅‖) times a (clogn)m...
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2018
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Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v466_n2_p1486_Marceca http://hdl.handle.net/20.500.12110/paper_0022247X_v466_n2_p1486_Marceca |
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paper:paper_0022247X_v466_n2_p1486_Marceca2023-06-08T14:48:01Z Some remarks on non-symmetric polarization Card-shuffling Main triangle projection Multilinear forms Polarization Polynomials Let P:Cn→C be an m-homogeneous polynomial given by P(x)=∑1≤j1≤…≤jm≤ncj1…jmxj1…xjm. Defant and Schlüters defined a non-symmetric associated m-form LP:(Cn)m→C by LP(x(1),…,x(m))=∑1≤j1≤…≤jm≤ncj1…jmxj1 (1)…xjm (m). They estimated the norm of LP on (Cn,‖⋅‖)m by the norm of P on (Cn,‖⋅‖) times a (clogn)m2 factor for every 1-unconditional norm ‖⋅‖ on Cn. A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schlüters’ argument) brings the constant term down to (cmlogn)m−1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than (clogn)m/2 when n≫m. Finally, the case of ℓp-norms ‖⋅‖p with 1≤p<2 is addressed. © 2018 Elsevier Inc. 2018 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v466_n2_p1486_Marceca http://hdl.handle.net/20.500.12110/paper_0022247X_v466_n2_p1486_Marceca |
institution |
Universidad de Buenos Aires |
institution_str |
I-28 |
repository_str |
R-134 |
collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
topic |
Card-shuffling Main triangle projection Multilinear forms Polarization Polynomials |
spellingShingle |
Card-shuffling Main triangle projection Multilinear forms Polarization Polynomials Some remarks on non-symmetric polarization |
topic_facet |
Card-shuffling Main triangle projection Multilinear forms Polarization Polynomials |
description |
Let P:Cn→C be an m-homogeneous polynomial given by P(x)=∑1≤j1≤…≤jm≤ncj1…jmxj1…xjm. Defant and Schlüters defined a non-symmetric associated m-form LP:(Cn)m→C by LP(x(1),…,x(m))=∑1≤j1≤…≤jm≤ncj1…jmxj1 (1)…xjm (m). They estimated the norm of LP on (Cn,‖⋅‖)m by the norm of P on (Cn,‖⋅‖) times a (clogn)m2 factor for every 1-unconditional norm ‖⋅‖ on Cn. A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schlüters’ argument) brings the constant term down to (cmlogn)m−1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than (clogn)m/2 when n≫m. Finally, the case of ℓp-norms ‖⋅‖p with 1≤p<2 is addressed. © 2018 Elsevier Inc. |
title |
Some remarks on non-symmetric polarization |
title_short |
Some remarks on non-symmetric polarization |
title_full |
Some remarks on non-symmetric polarization |
title_fullStr |
Some remarks on non-symmetric polarization |
title_full_unstemmed |
Some remarks on non-symmetric polarization |
title_sort |
some remarks on non-symmetric polarization |
publishDate |
2018 |
url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v466_n2_p1486_Marceca http://hdl.handle.net/20.500.12110/paper_0022247X_v466_n2_p1486_Marceca |
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1768543689074802688 |