Convergence and Boundedness of Cascade Algorithm in Besov Spaces and Triebel-lizorkin Spaces Running Title Convergence and Boundedness of Cascade Algorithm

In this paper, by introducing characteristic polynomial of a cascade algorithm and joint spectral radius on a nitely dimensional space, we give complete characterization of the rate of convergence and increment of the cascade algorithm in Besov spaces and TriebelLizorkin spaces. Also moment conditions of the initial distribution and the re nable distribution in the cascade algorithm, close relationship between regularity of the re nable distribution and convergence and boundedness of the cascade algorithm, and application of the characterization to nonhomogeneous re nement equations are studied. From our results, we see that the initial and the re nable distribution of the cascade algorithm satisfy less moment conditions for the boundedness of the cascade algorithm than for the convergence of the cascade algorithm, and for 0 < p < 1 than for p 1. It is observed that the convergence and boundedness of the cascade algorithm are equivalent to each other under certain restriction on the indices of regularity of function space and the rate of convergence of the cascade algorithm, and certain assumptions on the re nable distribution. AMS Subject Classi cation 42C15, 40A30, 46E35, 39B12 3