Existence of multiple refinable distributions (Q1371326)

The matrix refinement equation \[ \varphi= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha)\varphi(2\cdot-\alpha)\tag{1} \] in which \(a(\alpha)\) is a sequence of \(r\times r\) complex matrices and \(\varphi= (\varphi_1,\dots,\varphi_r)^T\) is a vector of distributions on \(\mathbb{R}^s\), is studied. One of the main results reads as follows. Theorem. Let \(\{a(\alpha)\}_{\alpha\in\mathbb{Z}^s}\) be a finitely supported sequence of \(2\times 2\) complex matrices. Assume that \(2^n\) is not an eigenvalue of \(M= 2^{-s}\sum_{\alpha\in \mathbb{Z}^s}a(\alpha)\) for any \(n\in\mathbb{N}\). Then (1) has a solution \(\varphi= (\varphi_1,\varphi_2)^T\), a vector of compactly supported distributions on \(\mathbb{R}^s\) such that \(\widehat\varphi(0)\neq 0\), if and only if the matrix \(M\) has an eigenvalue 1. In this case, \(\widehat\varphi(0)\) is an eigenvector of \(M\) associated with the eigenvalue 1. In the above statement \(\widehat\varphi(0)\) denotes the Fourier transform of \(\varphi\) at the point \(0\).