Riesz bases of exponentials on multiband spectra (Q2845737)

Suppose that \(S\subset\mathbb R^n\) is a bounded and measurable set. A classical question in harmonic analysis concerns the existence of Riesz bases of exponentials NEWLINE\[NEWLINEE(\Lambda) = \{\exp{2 \pi i \lambda x}, \quad \lambda \in \Lambda \}, NEWLINE\]NEWLINE for the space \(L^2(S)\), where \(\Lambda \subset {\mathbb R^n}\) is a discrete set.NEWLINENEWLINEOne very special case of a Riesz basis occurs when the exponentials can be chosen to be orthogonal and complete for \(L^2(S)\). For instance, if \(S= (0,1)^n\), then one can take \(\Lambda = {\mathbb Z^n}\) and obtain such an orthogonal basis of exponentials. This situation, where an orthogonal basis of exponentials exists, is not always guaranteed though and many ``standard'' domains do not have such a basis (e.g., the ball). This question relates to the so-called Fuglede or Spectral Set Conjecture [\textit{B. Fuglede}, J. Funct. Anal. 16, 101--121 (1974; Zbl 0279.47014)], claiming that for a set \(S\) to have such an orthogonal basis it is necessary and sufficient that it can tile space by translations (the conjecture was proved to be false in dimension 3 and higher).NEWLINENEWLINEThe existence of a Riesz basis for a domain \(S\) is an easier problem and, in dimension \(n=1\), it is known to be true when \(S \subset {\mathbb R}\) is an interval or is \(S\) is a finite union of disjoint intervals with commensurable lengths or the union of two general intervals.NEWLINENEWLINEThis paper shows that the result still holds true if \(S\) is the union of finitely many disjouint intervals on \({\mathbb R}\). In this case, the discrete set \(\Lambda\) for which there exists a Riesz bases of exponentials is obtained using quasicrystals, a method introduced by \textit{B. Matei} and \textit{Y. Meyer} to construct sampling and interpolating sets for Paley-Wiener spaces [C. R., Math., Acad. Sci. Paris 346, No. 23--24, 1235--1238 (2008; Zbl 1154.42006)].