Heisenberg Uniqueness Pair Research Articles

Heisenberg Uniqueness Pairs for the Finitely Many Parallel Lines with an Irregular Gap

Heisenberg Uniqueness Pairs for the Finitely Many Parallel Lines with an Irregular Gap

A note on discrete Heisenberg uniqueness pairs for the parabola

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.

Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on Cn. We derive that spheres as well as non-harmonic cones are determining sets for the spectral projections of the finite measure supported on the unit sphere. Further, we prove that if the Fourier transform of a finitely supported function on step two nilpotent Lie group is of arbitrary finite rank, then the function must be zero. The latter result correlates to the annihilating pair for the Weyl transform.

Müntz-Szász type theorems for the density of the span of powers of functions

The aim of this paper is to establish density properties in Lp spaces of the span of powers of functions {ψλ:λ∈Λ}, Λ⊂N in the spirit of the Müntz-Szász Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers {ψλ,ψλeiαt:λ∈Λ}. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines {cosλ⁡(t−θ1),cosλ⁡(t−θ2):λ∈Λ}. Under some arithmetic restrictions on θ1−θ2, we show that density is equivalent to a Müntz-Szász condition on Λ and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

Heisenberg uniqueness pairs on the Euclidean spaces and the motion group

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.

From Heisenberg uniqueness pairs to properties of the Helmholtz and Laplace equations

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of Rd agree on two intersecting d−1-dimensional submanifolds in generic position, then they agree everywhere.

Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on $${\mathbb {R}}^n$$ R n

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on $${\mathbb {R}}^n$$ . We derive that $$\left( S^2, \text { paraboloid}\right) $$ and $$\left( S^2, \text { geodesic of } S_r(o)\right) $$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in $${\mathbb {R}}^3$$ . Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.

A dynamical system approach to Heisenberg uniqueness pairs

Let Λ be a set of lines in R2 that intersect at the origin. For a smooth curve Γ ⊂ R2, we denote by AC(Γ) the subset of finite measures on Γ that are absolutely continuous with respect to arc length on Γ. For μ ∈ AC(Γ), $$\hat \mu $$ denotes the Fourier transform of μ. Following Hedenmalm and Montes-Rodríguez, we say that (Γ,Λ) is a Heisenberg uniqueness pair if μ ∈ AC(Γ) is such that $$\hat \mu $$ = 0 on Λ implies μ = 0. The aim of this paper is to provide new tools to establish this property. To do so, we reformulate the fact that $$\hat \mu $$ vanishes on Λ in terms of an invariance property of μ induced by Λ. This leads us to a dynamical system on Γ generated by Λ. In many cases, the investigation of this dynamical system allows us to establish that (Γ,Λ) is a Heisenberg uniqueness pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also gives a better geometric intuition as to why (Γ,Λ) is a Heisenberg uniqueness pair. As a side result, we also give the first instance of a positive result in the classical Cramér-Wold theorem where finitely many projections suffice to characterize a measure (under strong support constraints).

Heisenberg uniqueness pairs corresponding to a finite number of parallel lines

In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R2 so that (Γ,Λ) becomes a HUP.

Heisenberg uniqueness pairs for some algebraic curves and surfaces

Let $X(\Gamma)$ be the space of all finite Borel measure $\mu$ in $\mathbb R^2$ which is supported on the curve $\Gamma$ and absolutely continuous with respect to the arc length of $\Gamma$. For $\Lambda\subset\mathbb R^2,$ the pair $\left(\Gamma, \Lambda\right)$ is called a Heisenberg uniqueness pair for $X(\Gamma)$ if any $\mu\in X(\Gamma)$ satisfies $\hat\mu\vert_\Lambda=0,$ implies $\mu=0.$ We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the determining sets $\Lambda$ for $X(\Gamma)$ depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of certain trigonometric polynomials.

THE CRAMÉR–WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

Heisenberg uniqueness pairs for some algebraic curves in the plane

A Heisenberg uniqueness pair is a pair (Γ,Λ), where Γ is a curve and Λ is a set in R2 such that whenever a finite Borel measure μ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfies μˆ|Λ=0, then it is identically 0. In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.

Perron-Frobenius operators and the Klein-Gordon equation

For a smooth curve \Gamma and a set \Lambda in the plane \mathbb R^2 , let \mathrm{AC}(\Gamma;\Lambda) be the space of finite Borel measures in the plane supported on \Gamma , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on \Lambda . Following [12], we say that (\Gamma,\Lambda) is a Heisenberg uniqueness pair if \mathrm{AC}(\Gamma;\Lambda)=\{0\} . In the context of a hyperbola \Gamma , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets \Lambda of a collection of solutions to the Klein–Gordon equation. In this work, we mainly address the issue of finding the dimension of \mathrm{AC}(\Gamma;\Lambda) when it is non-zero. We will fix the curve \Gamma to be the hyperbola x_1x_2=1 , and the set \Lambda= \Lambda_{\alpha,\beta} to be the lattice-cross \Lambda_{\alpha,\beta}=\left(\alpha \mathbb Z\times\{0\}\right)\cup \left(\{0\}\times\beta \mathbb Z\right), where \alpha,\beta are positive reals. We will also consider \Gamma_+ , the branch of x_1x_2=1 where x_1>0 . In [12], it is shown that \mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta})=\{0\} if and only if \alpha\beta\le1 . Here, we show that for \alpha\beta>1 , we get a rather drastic “phase transition”: \mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta}) is infinite-dimensional whenever \alpha\beta>1 . It is shown in [13] that \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta})=\{0\} if and only if \alpha\beta<4 . Moreover, at the edge \alpha\beta=4 , the behavior is more exotic: the space \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is one-dimensional. Here, we show that the dimension of \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is infinite whenever \alpha\beta>4 . Dynamical systems, and more specifically Perron–Frobenius operators, will play a prominent role in the presentation.

Heisenberg uniqueness pairs in the plane. Three parallel lines

A Heisenberg uniqueness pair is a pair ( , ) , where is a curve in the plane and is a set in the plane, with the following property: any bounded Borel measure µ in the plane supported on , which is absolutely continuous with respect to arc length, and whose Fourier transform b µ vanishes on , must automatically be the zero measure. We characterize the Heisenberg uniqueness pairs for being three parallel lines = R◊{,, } with < < , ( )/( )2 N.

Heisenberg Uniqueness Pairs for the Parabola

Let Γ denote the parabola y=x2 in the plane. For some simple sets Λ in the plane we study the question whether (Γ,Λ) is a Heisenberg uniqueness pair. For example we shall consider the cases where Λ is a straight line or a union of two straight lines.

Heisenberg uniqueness pairs and the Klein-Gordon equation

A Heisenberg uniqueness pair (HUP) is a pair ( , ), where is a curve in the plane and is a set in the plane, with the following property: any finite Borel measure µ in the plane supported on , which is absolutely continuous with respect to arc length, and whose Fourier transformb vanishes on , must automatically be the zero measure. We prove that when is the hyperbola x1x2 = 1, and is the lattice-cross = ( Z ◊ {0}) ( ({0} ◊ Z), where , are positive reals, then ( , ) is an HUP if and only if 1; in this situation, the Fourier transform bµ of the measure solves the one-dimensional Klein-Gordon equation. Phrased di erently, we show that e i nt , e i n/t , n 2 Z, span a weak-star dense subspace in L 1 (R) if and only if 1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkho Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin

Let T denote the unit circle in the plane. For various simple sets Λ in the plane we shall study the question whether ( T , Λ ) is a Heisenberg uniqueness pair. For example, we shall consider the cases where Λ is a circle or a union of two straight lines. We shall also use a theorem of Beurling and Malliavin.