Abstract
In this article, we consider Heisenberg uniqueness pairs corresponding to the exponential curve and surfaces, paraboloid, and sphere. Further, we look for analogous results related to the Heisenberg uniqueness pair on the Euclidean motion group and related product group.
Highlights
The uncertainty principle states that both function and its Fourier transform cannot be localized simultaneously
As a version of the uncertainty principle, Hedenmalm and Montes-Rodríguez introduced the notion of the Heisenberg uniqueness pair
Let X (Γ) be the space of all finite complex-valued Borel measures in R2 which are supported on Γ and absolutely continuous with respect to the arc length measure on Γ, and for (ξ, η) ∈ R2, the Fourier transform of μ is defined by μ(ξ, η) = e−i π(x·ξ+y·η)dμ(x, y )
Summary
The uncertainty principle states that both function and its Fourier transform cannot be localized simultaneously (see [3,10,14]). In [11], Hedenmalm and Montes-Rodríguez propose the following problem: Let Γ be a finite disjoint union of smooth curves in R2 and Λ be any subset of R2. In [11], Hedenmalm and Montes-Rodríguez have shown that the pair (hyperbola, some latticecross) is a Heisenberg uniqueness pair. A weak-star dense subspace of L∞(R) has been constructed to solve the one-dimensional Klein–Gordon equation They characterize the Heisenberg uniqueness pairs corresponding to any two parallel lines. Babot [2] has given a characterization of the Heisenberg uniqueness pairs corresponding to a certain system of three parallel lines.
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