STABILITY AND ORTHONORMALITY OF MULTIVARIATE REFINABLE FUNCTIONS

This paper characterizes the stability and orthonormality of the shift s of a multidimensional (M,c) refinable function phi in terms of the e igenvalues and eigenvectors of the transition operator W-cau defined b y the autocorrelation c(au) of its refinement mask c, where M is an ar bitrary dilation matrix. Another consequence is that if the shifts of phi form a Riesz basis, then W-cau has a unique eigenvector of eigenva lue 1, and all of its other eigenvalues lie inside the unit circle. Th e general theory is applied to two-dimensional nonseparable (M,c) refi nable functions whose masks are constructed from Daubechies' conjugate quadrature filters.