Exponential bases for partitions of intervals

09/09/2021

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by   Goetz Pfander, et al.

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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∈ JΛ_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.