Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf= 0 for some integer n. Here△h= ∑Nj=1D2j is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G,Djf(x)=(δ)/(δ)xjf(x)+∑v∈R+kvf(x)-f(бvx)/vj,where kv is a multiplicity function on R and σv is the reflection with respect to the root v.We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+│x│2f1(x)+…+│x│2(n-1)fn-1(x),(A)x∈Ω,where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.
In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q(ψ1),Hu,v(ψ2)) for the values of p,q,u,v in three cases: (i) 0 ＜ p ≤ u ≤∞, 0 ＜ q ≤min(1,v) ≤∞. (ii) v= ∞,0 ＜ p ≤ u ≤∞, 1 ≤ u,q ≤∞. (iii) 1 ≤ v ≤ 2 ≤ q ≤∞, and 0 ＜ p ≤ u ≤∞ or 1 ≤ p,u ≤∞. The first case extends the result of Blasco, Jevti(c), and Pavlovi(c) in one variable. The third case generalizes partly the results of Jevti(c), Jovanovi(c), and Wojtaszczyk to higher dimensions.
多项式近似为 holo-morphic 功能空格 X 在 C^n 在围住的对称的领域Ω上被学习，例如 Bloch 类型空格， Bergman 类型空格，强壮的空格，Ω代数学和 Lipschitz 空间。我们扩大古典杰克逊“ s 定理到几个复杂变量：E_ k ( f ， X )近似＜ ω ( 1/k ， f ， X )在哪儿 E_k ( f ， X )至多是由度的多项式的 f ∈ X 的最好的近似的偏差 k 关于X公制并且ω ( l/k ， f ， X )是连续性的相应模量。
