On stable refinable function vectors with arbitrary support

Refinable function vectors with arbitrary support are considered. In particular, necessary conditions for stability are given and a characterization of the symbol associated with a stable refinable function vector in terms of the transfer operator is provided: this is a generalization of Gundy’s theorem to the vector case. The proof adapts the tools provided in [S. Saliani, On stability and orthogonality of refinable functions, Appl. Comput. Harmon. Anal. 21 (2006) 254–261]. Though complications arise from noncommuting matrix products, the fundamental ideas are the same.