Stability and orthonormality of multivariate refinable functions

Abstract

This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M, c) refinable function φ in terms of the eigenvalues and eigenvectors of the transition operator Wcau defined by the autocorrelation cau of its refinement mask c, where M is an arbitrary dilation matrix. Another consequence is that if the shifts of φ form a Riesz basis, then Wcau has a unique eigenvector of eigenvalue 1, and all of its other eigenvalues lie inside the unit circle. The general theory is applied to two-dimensional nonseparable (M, c) refinable functions whose masks are constructed from Daubechies' conjugate quadrature filters.