A set with no Riesz basis of exponentials

TL;DR

This paper demonstrates the existence of a bounded set in R for which no exponential system forms a Riesz basis, and provides a lower bound for the Riesz constant in the 2D disk, advancing understanding of basis limitations.

Contribution

It introduces a specific bounded set lacking a Riesz basis of exponentials and establishes a lower bound for Riesz constants in two dimensions, revealing fundamental constraints.

Findings

Existence of a bounded set with no Riesz basis of exponentials

Lower bound for Riesz constant in 2D disk

Constraints on exponential basis constructions

Abstract

We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two dimensional disk.