Régularisation spectrale et propriétés métriques des moyennes mobiles

Abstract

We develop a spectral regularization technique for moving averages$B_n^{U,\phi } = \frac{1}{n}\sum\nolimits_{j = \phi (n)}^{\phi (n) + n - 1} {U^j }$, where ϕ is a nondecreasing map andU: H→H is a contraction of a Hilbert space (H, ‖·‖). We obtain a spectral regularization inequality which allows one to evaluate efficiently the increments ‖B ^U _m , ϕ (f)−B ^U _n , ϕ (f)‖,f∈H, by means of$\hat \mu [1/m,1/n]$ where$\hat \mu$ is a properly regularized version of the spectral measure off with respect toU. We apply this inequality to an investigation of metric properties of the sets of moving averages {B ^{U, ϕ} _n (f), n ∈N} with fixedf∈H andN⊂ N. In particular, we obtain estimates of the associated covering numbers as well as of the related Littlewood-Paley-type square functions. This work extends our previous results concerning the case of classical averages (ϕ(n)=0). Since it is well-known that the structure of general moving averages is more complicated, there is no surprise that the general results we obtain are sometimes less complete than their classical counterparts and need suitable moment assumptions on the spectral measure (depending on the growth of the shift function ϕ). Nevertheless, when applied to the classical situation, our estimates still yield optimal bounds.