Geodesic
Discrete uniqueness sets for functions with spectral gaps
Sbornik. Mathematics, Tome 208 (2017) no. 6, pp. 863-877

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It is well known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets. We show that the same is true for a much wider range of spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having ‘periodic gaps’. The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that uniformly discrete sets $\Lambda$ satisfy a strong non-uniqueness property: every discrete function $c(\lambda)\in l^2(\Lambda)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$. Bibliography: 9 titles.
Keywords: spectral gap, discrete uniqueness set, Sobolev space.
Mots-clés : Fourier transform 
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     title = {Discrete uniqueness sets for functions with spectral gaps},
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Alexander Olevskii; Alexander Ulanovskii. Discrete uniqueness sets for functions with spectral gaps. Sbornik. Mathematics, Tome 208 (2017) no. 6, pp. 863-877. http://geodesic.mathdoc.fr/item/SM_2017_208_6_a5/