On Fourier asymptotics of a generalized Cantor measure

Bérenger Akon Kpata, Ibrahim Fofana, and Konin Koua. "On Fourier asymptotics of a generalized Cantor measure." Colloquium Mathematicae 119.1 (2010): 109-122. <http://eudml.org/doc/283841>.

@article{BérengerAkonKpata2010,
abstract = {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.},
author = {Bérenger Akon Kpata, Ibrahim Fofana, Konin Koua},
journal = {Colloquium Mathematicae},
language = {eng},
number = {1},
pages = {109-122},
title = {On Fourier asymptotics of a generalized Cantor measure},
url = {http://eudml.org/doc/283841},
volume = {119},
year = {2010},
}

TY - JOUR
AU - Bérenger Akon Kpata
AU - Ibrahim Fofana
AU - Konin Koua
TI - On Fourier asymptotics of a generalized Cantor measure
JO - Colloquium Mathematicae
PY - 2010
VL - 119
IS - 1
SP - 109
EP - 122
AB - 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.
LA - eng
UR - http://eudml.org/doc/283841
ER -