Multiple Refinable Distributions: Existence and Factorization

We consider the existence of compactly supported distribution solutions ϕ:= (ϕ₁,...,ϕ_r)^T to matrix refinement equations of the form

where a is a finitely supported sequence of r x r matrices called the refinement mask. Such multiple refinable distributions occur naturally in the study of multiple wavelets. A necessary condition for the existence of ϕ with https://www.w3.org/1998/Math/MathML"> ϕ ^ ( 0 ) ≠ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math1127.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is that the matrix Σa(α)/2 has an eigenvalue 1. However, this condition is not sufficient, which is different from the scalar case. In this paper we provide a necessary and sufficient condition for the existence of a solution ϕ subject to https://www.w3.org/1998/Math/MathML"> ϕ ^ ( 0 ) ≠ 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math1128.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Such a solution can always be obtained through some infinite matrix product. Our condition is totally based on the refinement mask. A general factorization of the refinement mask is presented in the univariate case, which gives a way of constructing multiple refinable distributions.