A splitting problem for unconditional bases of complex exponentials (Q2782550)

The main result of this paper is the following. NEWLINENEWLINENEWLINETheorem. Let \({\mathcal E}(\Lambda)\) be an unconditional basis in \(L^2(-\pi, \pi)\). Then for each \(a\) with \(0 < a <1\) there exists a splitting NEWLINE\[NEWLINE \Lambda = {\Lambda}' + {\Lambda}'', \qquad {\Lambda}' \cap {\Lambda}''= \emptyset NEWLINE\]NEWLINE such that \({\mathcal E}({\Lambda}')\) and \({\mathcal E} ({\Lambda}'')\) are unconditional bases in \(L^2(-a \pi, a \pi)\) and \(L^2(-(1-a) \pi, (1-a) \pi)\), respectively.NEWLINENEWLINENEWLINEThe authors also also give an example showing that the converse is not necessarily true.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].