Characterization of principal shift-invariant spaces on Vilenkin group

Abstract Translation generated spaces of square-integrable functions on the Vilenkin group are studied using a C-bracket. Principal shift-invariant spaces are characterized via the periodization function and the C-bracket. Conditions for semi-orthogonal Parseval wavelets are established, leading to a generalized multiresolution analysis (GMRA) of the semi-orthogonal Parseval frame. Examples are provided to support and illustrate the theoretical results.