Orthogonality criteria for compactly supported refinable functions and refinable function vectors

Abstract

A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(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 {φj(x−α):α∈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 multiwavelet 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.