THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

96 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesquantity M 3 (β)(on the vertical axis) as a function of β ∈ (0, 1/2].2.01.91.81.71.61.51.41.31.21.11.00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50We can see that, for β ∈ (0, 1/2], 1.06 ≤ M 3 (β) ≤ 2 and then it implies that1.06 ≤ 1.06( ) βc2 (β)c 1 (β)2β+1≤ M2 (β) ≤ 2(c2 (β)c 1 (β)) β2β+1.The following sections are devoted to preliminary results and to the proofs of the theorems.6.3 Some preliminary resultsIn the following, D i , with i = 1, 2, . . ., denote positive constants, except otherwise mentioned.These constants can depend on β ∈ B but we do not indicate explicitly thedependence on β. This does not have consequences on the proofs since B is a finite set.The quantity η j (β) satisfies the following lemma which will be proved in Section 6.8.Lemma 6.2. For β ∈ B and j ∈ {1, 2}, we haveη j (β) + jψ n(β)λ j (β)2β + 1= M j (β)ψ n (β),where M 1 (β) = 1. Moreover, for β ∈ B, we haveη 3 (β) + 2ψ n(β)λ 3 (β)2β + 1= M 3 (β)ψ n (β).We have the following results for the families of estimators ( ˆf β,1 ) β∈B , ( ˆf β,2 ) β∈B and( ˆf β,3 ) β∈B .Proposition 6.1. For β ∈ B and j ∈ {1, 2, 3}, we havesupf∈Σ(β,L)‖b β,j (·, f)‖ ∞ ≤ ψ n(β)λ j (β).2β + 1