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) = [φ<inf>l</inf>(x), . . . , φ<inf>r</inf>(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<inf>j</inf>(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.