| Abstract: | In this paper, the L2-convergence of vector cascade algorithm (a very important concept in the wavelet theory) is characterized in terms of the behavior of the given filter matrix H(ξ) and its associated transition operator TH. More precisely, it is shown that the cascade algorithm defined by H(ξ) converges in L2(Rd) if and only if H(ξ) satisfies Fundamental Condition (Condition F, a restriction on its values on the vertices of [0, π]d), and satisfies either Condition E(1 is a simple eigenvalue and the others have modulees less than one) or Condition [(E)tilde]tilde E . Also, we describe a class of functions which can be chosen as initial functions. |
