REFINEMENT EQUATIONS AND CORRESPONDING LINEAR OPERATORS

Abstract

Refinement equations of the type [Formula: see text] play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T0, T1 constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T0 and T1, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.