In this paper we shall study the initial functions in a cascade algorithm in a Sobolev space. By investigating the mutual relations among the initial functions in a cascade algorithm, we are able to study in a relatively unified approach several questions related to cascade algorithms and subdivision schemes in a Sobolev space $W_p^k(\RR^s)$ ($1\le p \le \infty, k\in \NN\cup \{0\}$) such as convergence, error estimate and convergence rate of cascade algorithms and subdivision schemes in a Sobolev space with a general isotropic dilation matrix. The approach in this paper can also be used to study vector cascade algorithms and vector subdivision schemes with a dilation matrix in a multidimensional space.