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Chapitre D'ouvrage Année : 2010

Integral Concentration of idempotent trigonometric polynomials with gaps

Résumé

We prove that for all p>1/2 there exists a constant $\gamma_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $\gamma<\gamma_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > \gamma \int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $\gamma_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $\gamma_p=1$ 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

Dates et versions

hal-00184959 , version 1 (04-11-2007)

Identifiants

Citer

Aline Bonami, Szilárd Gy. Révész. Integral Concentration of idempotent trigonometric polynomials with gaps. Recent Developments in Fractals and Related Fields, Birkhaüser, pp.107--129, 2010. ⟨hal-00184959⟩
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