Exponential bases for partitions of intervals
Abstract For a partition of [0, 1] into intervals I 1,…, I n we prove the existence of a partition of
Z into Λ 1,…, Λ n such that the complex exponential functions with frequencies in Λ k form a
Riesz basis for L 2 (I k), and furthermore, that for any J⊆{1, 2,…, n}, the exponential
functions with frequencies in⋃ j∈ J Λ j form a Riesz basis for L 2 (I) for any interval I with
length| I|=∑ j∈ J| I j|. 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 …
Z into Λ 1,…, Λ n such that the complex exponential functions with frequencies in Λ k form a
Riesz basis for L 2 (I k), and furthermore, that for any J⊆{1, 2,…, n}, the exponential
functions with frequencies in⋃ j∈ J Λ j form a Riesz basis for L 2 (I) for any interval I with
length| I|=∑ j∈ J| I j|. 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 …
Abstract
Abstract For a partition of [0, 1] into intervals I 1,…, I n we prove the existence of a partition of Z into Λ 1,…, Λ n such that the complex exponential functions with frequencies in Λ k form a Riesz basis for L 2 (I k), and furthermore, that for any J⊆{1, 2,…, n}, the exponential functions with frequencies in⋃ j∈ J Λ j form a Riesz basis for L 2 (I) for any interval I with length| I|=∑ j∈ J| I j|. 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.
Elsevier