Refinable functions for dilation families

Abstract

We consider a family of d × d matrices W e indexed by e ∈ E where (E, μ) is a probability space and some natural conditions for the family (W e ) e ∈ E are satisfied. The aim of this paper is to develop a theory of continuous, compactly supported functions $\varphi: {{\mathbb R}}^d \to {\mathbb{C}}$ which satisfy a refinement equation of the form $$\varphi (x) = \int_E \sum\limits_{\alpha \in {{\mathbb Z}}^d} a_e(\alpha)\varphi\left(W_e x - \alpha\right) d\mu(e)$$ for a family of filters $a_e : {{\mathbb Z}}^d \to {\mathbb{C}}$ also indexed by e ∈ E. One of the main results is an explicit construction of such functions for any reasonable family (W e ) e ∈ E . We apply these facts to construct scaling functions for a number of affine systems with composite dilation, most notably for shearlet system