Local linear independence of refinable vectors of functions

This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions φ = (φ₁, . . . , φ_r)^T ∈ (C(ℝ^s))^r is said to be refinable if it satisfies the vector refinement equation φ(x) = ∑_α∈ℤs a(α)φ(2x - α), where a is a finitely supported sequence of r x r matrices called the refinement mask. A complete characterization for the local linear independence of the shifts of φ1, . . . , φ_r is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets.