2023/09/26 by Lai, Chun-Kit, Sheynis, Alexander · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2309.14625
Let G be a closed subgroup of \mathbb Rd and let ν be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group G⊥. We form a multi-tiling measure μ= μ1+...+μN where μi is translationally equivalent to ν and different μi and μj have essentially disjoint support. We obtain some necessary and sufficient conditions for μ 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× \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× \0\. This rules out a natural candidate of potential Riesz basis for the square boundary.