Integral Concentration of idempotent trigonometric polynomials with gaps
| dc.creator | Bonami, Aline | |
| dc.creator | Révész, Szilárd Gy. | |
| dc.date | 2007-07-20 | |
| dc.date | 2008-10-16 | |
| dc.date.accessioned | 2026-07-07T10:10:06Z | |
| dc.date.available | 2026-07-07T10:10:06Z | |
| dc.description | We prove that for all p>1/2 there exists a constant $γ_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $γ<γ_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > γ\int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $γ_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $γ_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows striking differences with the case p=2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener. We find sharper results for $0<p\leq 1$ when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones. | |
| dc.description | 43 pages; to appear in Amer. J. Math | |
| dc.identifier | https://arxiv.org/abs/0707.3023 | |
| dc.identifier | http://arxiv.org/abs/0707.3023 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/171519 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 42A05; 42A55 | |
| dc.title | Integral Concentration of idempotent trigonometric polynomials with gaps | |
| dc.type | text |