Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x): ℝⁿ → ℝ or, more generally, a refinable function vector Φ(x) = [φ₁(x),…, φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if(φ_j(x-α): α ε ℤⁿ, 1 ≤ j ≤ r] form an orthogonal set of functions in L² (ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multi-wavelet bases of L² (ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.