【摘要】 研究细分方程φ(x)=∑α∈Zsa(α)φ(Mx-α),x∈Rs,其中向量值函数φ=(φ1,φ2,…,φr)T∈(Lp(Rs))r(1≤p≤∞),a(α)是具有有限长的r×r矩阵值序列,称为面具,M是一个s×s整数矩阵且满足li mn→∞M-n=0.定义φn∶=Qanφ0,n=1,2,…,其中,Qaφ(x)=∑α∈Zsa(α)φ(Mx-α),φ∈(Lp(Rs))r,函数列{φn}n≥0称为细分格式或级联序列.利用由M,a(α)以及集合E生成的有限线性算子的联合谱半径来刻画与a,φ0,M有关的细分格式{φn}n≥0的收敛阶,其中集合E表示含0的商群Zs/MZs的不同代表元.更多还原

【Abstract】 A refinement equation of the form φ（x）=∑α∈Z^sa（α）φ（Mx-α）,x∈R^s is concerned,where the vector og functions φ=（φ₁,φ₂,…,φ_r）^T∈（L_p（R^s））^r（1≤p≤∞） is unknown,a is a finitely supported sequence of r×r matrices called refinement mask,and M is an s×s integer matrix so that（lim）n→∞M^-n=0.The cascade algorithms is associated with a,M generates a sequence φ_n=Q_aⁿφ₀,n=1,2,… by Q_aφ（x）=∑α∈Z^sa（α）φ（Mxα）,φ∈（L_p（R^s））^r,from a starting vector of function φ₀ in（L_p（R^s））^r（1≤p≤∞）.The aim is to give a characterization of convergence rates of the cascade algorithms {φ_n} in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by a and the set E,where the set E is a complete set of representatives of the distinct cosets of the quotient group Z^s/MZ^s containing 0.更多还原