Abstract
For a partition of [0,1] into intervals I1,…,In we prove the existence of a partition of Z into Λ1,…,Λn such that the complex exponential functions with frequencies in Λk form a Riesz basis for L2(Ik), and furthermore, that for any J⊆{1,2,…,n}, the exponential functions with frequencies in ⋃j∈JΛj form a Riesz basis for L2(I) for any interval I with length |I|=∑j∈J|Ij|. The construction extends to infinite partitions of [0,1], but with size limitations on the subsets J⊆Z; it combines the ergodic properties of subsequences of Z known as Beatty-Fraenkel sequences with a theorem of Avdonin on exponential Riesz bases.
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