On Fourier asymptotics of a generalized Cantor measure

Bérenger Akon Kpata ; Ibrahim Fofana ; Konin Koua

Colloquium Mathematicae, Tome 120 (2010), p. 109-122 / Harvested from The Polish Digital Mathematics Library

Résumé

Let d be a positive integer and μ a generalized Cantor measure satisfying $μ = ∑_{j = 1}^{m} a_{j}μ∘S_{j}^{-1}$, where $0 < a_{j} < 1$, $∑_{j = 1}^{m}a_{j} = 1$, $S_{j} = ρR + b_{j}$ with 0 < ρ < 1 and R an orthogonal transformation of $ℝ^{d}$. Then ⎧1 < p ≤ 2 ⇒ ⎨$sup_{r>0} r^{d(1/α^{\prime }-1/p^{\prime })} (∫_{J_{x}^{r}} |μ̂(y)|^{p^{\prime }}dy)^{1/p^{\prime }} ≤ D₁ρ^{-d/α^{\prime }}$, $x ∈ ℝ^{d}$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$, $where $J_{x}^{r} = ∏_{i=1}^{d} (x_{i} - r/2,x_{i} + r/2)$, α’ is defined by $ρ^{d/α^{\prime }} = (∑_{j=1}^{m} a_{j}^{p})^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

Détail

Publié le : 2010-01-01

Zbl 1189.42002

EUDML-ID : urn:eudml:doc:283841