There are no Riesz bases of exponentials in balls and triangles
Abstract
We prove that L2(Ω) admits no Riesz basis of exponentials when Ω is a ball in Rn, n≥2, or a nondegenerate triangle in R2. The ball result extends to every nonempty bounded convex open set with C2 boundary in Rn, n≥2. 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.
Related
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- B-Frames, B-Riesz bases, and Their tensor products
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BibTeX
@misc{ortegacerda2026there,
title = {{There are no Riesz bases of exponentials in balls and triangles}},
author = {Joaquim Ortega-Cerdà},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18426},
eprint = {2609.18426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18426},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - There are no Riesz bases of exponentials in balls and triangles
AU - Joaquim Ortega-Cerdà
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.18426
UR - https://arxiv.org/abs/2609.18426
AB - We prove that L^2(Ω) admits no Riesz basis of exponentials when Ω is a ball in R^n, n≥2, 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≥2. 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.
ER -