Abstract
We discuss on Heisenberg uniqueness pairs for the parabola given by discrete sequences along straight lines. Our method consists in linking the problem at hand with recent uniqueness results for the Fourier transform.
Highlights
For an algebraic curve Γ ⊂ R2 and a set Λ ⊂ R2, we say that the pair (Γ, Λ) is a uniqueness pair if whenever μ is a borel complex measure on Γ, absolutely continuous with respect to the arc-length measure on Γ, μ ∈ L1(Γ) and μ|Λ = 0, μ ≡ 0
We define the Fourier transform1 of a measure μ on R2 by μ(ξ, η) = e−2πi(ξx+ηy) dμ(x, y). This concept is inspired in the classical Heisenberg uncertainty principle for the Fourier transform, but it has been explored in more depth since the pioneering work of Hedenmalm and Montes-Rodriguez [5]
There, the authors analyze some classical cases of Heisenberg uniqueness pairs with Γ being a conic section
Summary
There, the authors analyze some classical cases of Heisenberg uniqueness pairs with Γ being a conic section Their main results concern the hyperbola Γ = {(x, y) ∈ R2 : xy = 1} and lattice-crosses of the form Λα,β = (αZ × {0}) ∪ ({0} × βZ), where they prove (Γ, Λα,β) is a uniqueness pair if and only if αβ ≤ 1. We aim to look at this concept for the parabola case Γ = P and a set Λ consisting of discrete points along two or three lines This can be seen, for instance, as a natural complement to the results of Hedenmalm and Montes-Rodriguez [5], replacing the hyperbola with the parabola. The rest of this manuscript is devoted to the proof of Theorem 1, together with a few comments on its proof and possible generalizations
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