Riesz bases of exponentials for multi-tiling measures

Abstract

Let G be a closed subgroup of ${\mathbb {R}}^d$ and let $\nu$ be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group $G^{\perp }$. We form a multi-tiling measure $\mu = \mu _1+\cdots +\mu _N$ where $\mu _i$ is translationally equivalent to $\nu$ and different $\mu _i$ and $\mu _j$ have essentially disjoint support. We obtain some necessary and sufficient conditions for $\mu$ to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of $G = {\mathbb {Z}}\times {\mathbb {R}}$ and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups ${\mathbb {Z}}\times \{0\}$. This rules out a natural candidate of potential Riesz basis for the square boundary.