Bases of complex exponentials with restricted supports

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2023-05, Vol.521 (2), p.126917, Article 126917
Main Authors: Lee, Dae Gwan, Pfander, Götz E., Walnut, David
Format: Article
Language:English
Subjects:
Complex exponentials
Riesz bases
Spectrum
Support restriction
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Summary:The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if S1,…,SK⊂[0,1] are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of Z into sets Λ1,…,ΛK such that ⋃k=1K{e2πiλ(⋅)χSk:λ∈Λk} is a Riesz basis for L2[0,1]. Here, χS denotes the characteristic function of S.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126917