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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todo:paper_0022247X_v466_n2_p1486_Marceca2023-10-03T14:29:24Z Some remarks on non-symmetric polarization Marceca, F. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 |
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Card-shuffling Main triangle projection Multilinear forms Polarization Polynomials Marceca, F. 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. |
| format |
JOUR |
| author |
Marceca, F. |
| author_facet |
Marceca, F. |
| author_sort |
Marceca, F. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_0022247X_v466_n2_p1486_Marceca |
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AT marcecaf someremarksonnonsymmetricpolarization |
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