摘要
Let E be a Moran set on R1 associated with a bounded closed interval J and two sequences(nk)k =1 ^∞ and(Ck =(ck,j)j=1)^nk)k≥1. Let μ be the Moran measure on E associated with a sequence(Pk)k≥1 of positive probability vectors with Pk =(pk,j)j =1,^nk, k ≥ 1. We assume that k≥1 1≤j≤nk^inf min Ck,j>0,k≥1 1≤j≤nk^inf minCk,j>0,k≥1 1≤j≤nk ^inf min pk,j>0. For every n ≥ 1, let αn be αn n optimal set in the quantization for μ of order r ∈(0,∞) and{Pa(αn)}a∈α∈an an arbitrary Voronoi partition with respect to αn. We write Iα(α,μ):=∫Pα(αn)^d(x,αn)^τ dμ(x),α∈αn;J(αn,μ):=α∈αn^minⅠα(α,μ),J(αn,μ):=α∈αn^max Ⅰα(α,μ).We show that J(αn,μ),J(αn,μ) and en^r,r(μ)-en^r +1,r(μ) are of the same order as 1/n en^r ,r(μ), where en^r ,r(μ):=∫d(x,αn)^r dμ(x) is the nth quantization error for μ of order r. In particular, for the class of Moran measures on R1, our result shows that a weaker version of Gersho’s conjecture holds.
Let E be a Moran set on R1 associated with a bounded closed interval J and two sequences（nk）k∞=1 and(Ck =（ck,j）jnk =1)k≥1. Let μ be the Moran measure on E associated with a sequence（Pk）k≥1 of positive probability vectors with Pk =（pk,j）jnk =1, k ≥ 1. We assume that ■ For every n ≥ 1, let αn be an n optimal set in the quantization for μ of order r ∈（0, ∞） and{Pa（αn）}a∈αnan arbitrary Voronoi partition with respect to αn. We write ■ We show that ■（αn, μ）, ■（αn, μ） and enr,r（μ）-enr +1,r（μ） are of the same order as 1/n enr ,r（μ）, where enr ,r（μ） :=?d（x, αn）r dμ（x） is the nth quantization error for μ of order r. In particular, for the class of Moran measures on R1, our result shows that a weaker version of Gersho’s conjecture holds.
基金
supported by National Natural Science Foundation of China(Grant No.11571144)
