There are no Riesz bases of exponentials in balls and triangles

Abstract

We prove that $L^2(Ω)$ admits no Riesz basis of exponentials when $Ω$ is a ball in $\R^n$, $n\geq2$, or a nondegenerate triangle in $\R^2$. The ball result extends to every nonempty bounded convex open set with $C^2$ boundary in $\R^n$, $n\geq2$. Starting from a hypothetical Riesz basis, we use Beurling weak limits of translates of its frequency set to construct a stationary system. On the associated Hilbert space, we define spectral cutoff projections and a comparison family of orthogonal projections obtained from the synthesis operator. The distance between corresponding projections is uniformly bounded by a constant strictly smaller than one. The comparison family is continuous in operator norm, whereas the boundary geometry of $Ω$ produces two strong limits of the cutoff projections with strictly nested ranges. Both limiting projections would then be at distance less than one from the same comparison projection, which is impossible.