In the article [18], Hedenmalm and Montes-Rodr´ıguez have shown that the pair (hyperbola, some discrete set) is a special Heisenberg pair. Then, some of the unique Heisenberg pairs corresponding to the parabola were obtained by Sj¨olin [44].
Introduction
- HUPs corresponding to the spiral
- HUPs corresponding to the exponential curves
- HUPs corresponding to the circle
- HUPs corresponding to the hyperbola
In this section we work out some of the Heisenberg uniqueness pairs corresponding to the circle. In this section we work out some of the Heisenberg uniqueness pairs corresponding to the hyperbola.
HUPs corresponding to certain surfaces
Then (Γ,Λ) is a Heisenberg singularity pair if and only if Λ is parallel to the hyperplane xn+1 = 0. Since Λ is an affine hyperplane in Rn+1 of dimension n, we can assume that Λ is a linear subspace due to the invariant properties of HUP Rn+1, which can be treated as xn+1 =cx1, where c∈R or x1 = 0. Conversely, suppose that Λ is not parallel to the hyperplane xn+1 = 0. Consider a non-zero supporting odd function ψ ∈L1(R) together with the zero compact support function h∈L1(Rn−1).
C - Heisenberg uniqueness pair
In view of the injectivity of the Fourier transform onL1(R), for a pair (Γo,Λ) to be aC-HUP, it is necessary that at least one of the orthogonal projectionsπj(Λ); j = 1,2 on the axes must be dense in R. It would be an interesting question to obtain a sufficient condition for C-Heisenberg unique pairs corresponding to Γo. A characterization of the Heisenberg singularity pair corresponding to two parallel straight lines has been carried out by Hedenmalm and Montes-Rodr´ıguez [18].
Furthermore, Babot [5] has worked out an analogous result for a certain system of three parallel lines. In this chapter we prove a characterization of Heisenberg uniqueness pairs corresponding to a given system of four parallel lines.
Preliminaries and auxiliary results
Also, it is easy to verify that (Γ,Λ) is a HUP if and only if (Γ,£(Λ)) is a HUP, where £(Λ) denotes the closure of£(Λ) in R2. Considering the above facts, it is sufficient to work with the closed set Λ ⊂R2, which is 2-periodic with respect to the second variable. However, they can locally agree to the Fourier transform of functions in L1(R). Therefore, considering the condition ˆµ|Λ = 0, we can classify these related exponential functions. To describe the rest of the two partition sets, we will use the concept of a symmetric polynomial.
On the basis of structural properties of the dispensable sets, we observe that these sets essentially minimize the size of the projection Π(Λ). Now we can state our main result of this chapter on the Heisenberg unique pairs corresponding to the above described system of four parallel straight lines.
The main result
Therefore, for any p≥ 3, the set Πfp(Λ) is correctly included in Π(Λ).e Thus, an analogous result for a four-line problem compared to the three-line problem is still unsolved. The main idea behind these lemmas is to pull down an interval from some of the partition sets of the projection Π(Λ). Since ρ is a symmetric polynomial in a, b, c, according to the fundamental theorem of symmetric polynomials, ρ can be expressed as a polynomial in ej; j = 0,1,2.
According to the hypothesis, I∩Π3∗(Λ) is dense inI, there exists an interval Iξ¯⊂I containing ¯ξ, such that ρ can be continuously expanded on Iξ¯. Consequently, J∩Π3∗(Λ) is compact in J and therefore for ξ∈J there exists a series ξn ∈J∩Π3∗(Λ) such that ξn →ξ. So the corresponding image sequences ηj(n)∈Σξn ⊆ [0,2) will have convergent subsequences, for example ηj(nk), which converge to ηj; j = 0,1,2.
Proof of Theorem 3.3.1
Then we claim that the quantity Π(Λ) is dense in R. We note that this is possible if the dispensable quantities Πj∗(Λ); j = 1,2,3 interlace to each other, even though these sets are mutually disjoint. Now the remainder of the proof of Theorem 3.3.1 is a consequence of the following two lemmas, which provide the interlacing property of the dispensable sets Πj∗(Λ); j = TH. The next lemma is to deal with the situation that any interval J ⊂ Io cannot contain only the points of a pair of dispensable sets.
In light of Lemma 3.4.1, we have thus arrived at a contradiction to the assumption that Io cuts only the dispensable sets. Using Lemma 3.3.2, there exists an interval I0 ⊂ J such that I0 is contained in Π3∗(Λ)∪Π4(Λ), which contradicts the assumption that Io cuts only the dispensable sets.
Remarks and open problems
In this chapter we work out an analogue of Benedick's theorem for the Euclidean motion group M(n). We prove that if the Fourier transform of certain integrable functions is of finite rank, then the function must vanish identically. Further, we investigate the possibility of the Heisenberg uniqueness pairs for the Fourier transform on M(n) as well as on the product group G0 =Rn×K. In the latter case, we observed a one-to-one correspondence between the class of HUPs on Rn and the class of HUPs on G0.
Notation and preliminaries
Then it can be shown that an infinite-dimensional unitary irreducible representation of G is the restriction of πa,σ to H(K,Cdσ). Since the Plancherel measure μσ in ˆG can be expressed as dµσ =cnan−1da, where cn depends only on n, the corresponding Plancheral formula is given by. Next, we would need a concrete realization of the representations in ˆKM, which can be done in the following way.
In fact, the corresponding unit representation πl is in ˆKM. Furthermore, ˆKM can be identified, up to unitary equivalence, with the set {πl:l ∈Z+}. Furthermore, a homogeneous polynomial can be uniquely decomposed in terms of homogeneous harmonic polynomials, it follows that (4.2.6) holds.
Results on the Euclidean motion group M (n)
We further prove that the radial function on G can be determined by its Fourier transform at one point.
Some auxiliary results on compact group
Further, using the Peter-Weyl theorem, we prove that if g ∈ L1(K), then the operator W(g) has finite rank as long as g is a trigonometric polynomial. Then the operator W(g) is of finite rank if and only if g is a trigonometric polynomial on K. Therefore ˆϕj, q(δ) 6= 0 at most for finitely many δ ∈ K.ˆ Thus by the Peter-Weyl theorem, derive vi that ϕj, q is a trigonometric polynomial.
Conversely, suppose that g is a trigonometric polynomial, then without loss of generality we can assume that g =ϕδij. Note that using the inverse Fourier transform on both sides of (4.4.2), we can assume that f is a trigonometric polynomial as long as f(aˆ o, σ) is a finite-rank operator for some ao ∈R+ and σ ∈M. ˆ.
Some results on the Heisenberg uniqueness pairs
We would like to mention the necessity of the non-vanishing conditions of the Fourier coefficients in the above problem. In other words, the support of an integrable function and its Fourier transform cannot be of finite size simultaneously. In the article [31], an analogous result on the Heisenberg group is worked out for compactly supported partial functions in terms of the finite order of the Fourier transform of the function.
In this article, we explore results analogous to the Amrein-Berthier and Benedicks theorem on the Heisenberg motion group and second nilpotent Lie groups. However, it would be reasonable to consider the case when the spectrum of the Fourier transform of an integrable function will rest on an uncountably thin set.
Preliminaries on the Heisenberg motion group
Let φλα(x) = |λ|n4φα(p . |λ|x); α ∈ Zn+, where φα are Hermitian functions on Rn. Then φλα are eigenfunctions of Hλ with eigenvalue (2|α|+n)|λ|. Therefore, the input functions Eαβλ of the representation πλ of the eigenfunction under the -Laplacian L are satisfactory. The Heisenberg motion group G is a group of isometries Hn that leaves an invariant sub-Laplacian L. Since the action of the unitary group K =U(n) defines an automorphism group on Hnviak·(z, t) = (kz, t ), where k ∈K , the group G can be expressed as a semi-direct product of Hn and K. A function with operator value µλ can be thought of as a unitary representation of the group K on L2(Rn) and is known as a metaplectic representation.
Let (σ,Hσ) be an irreducible unitary representation of K andHσ = span{eσj : 1≤j ≤dσ}. Fork ∈K,the matrix coefficients of the representation σ∈K,ˆ are defined. So, in light of the above argument, we denote the partial dual of the group G by G0 ∼=R∗×K.ˆ.
Uniqueness results on the Heisenberg motion group
Using the scale argument it is sufficient to prove these results for the case λ= 1. ii) Then let dg=dzdk. It follows from the orthogonality of the special Hermite functions φαβ together with the identity (5.3.1). Next we recall the Peter-Weyl theorem which is crucial for the proof of Theorem 5.3.3.
According to Theorem 5.3.3, we prove the following analogous result for the Fourier-Wigner transform. Since ¯τ is finitely supported in the variable Cn, it follows from Proposition 5.3.5 that τσ = 0 whenever σ ∈ K.ˆ According to Plancherel's formula for the Weyl transform, as mentioned in Proposition 5.3.2, we conclude, that F = 0.
Preliminaries on step two nilpotent group
We prove the result for λ = 1 and the general case will follow through the scaling argument. Let G be connected, simply connected Lie group with real step two nilpotent Lie algebra g. Then g has the orthogonal decomposition g=b⊕z, where z is the center of g. Since mω is invariant under the skew-symmetric bilinear form Bω, it follows that the dimension of mω is even.
Next, we briefly describe the irreducible representation of the M´etivier group G which can be parameterized by Λ.
Uniqueness results on step two nilpotent group
If the Fourier transform of a compactly supported function f onHnnU(n) (or second-step nilpotent Lie groups) reduces to the space of compact operators, then f may be zero. However, it would be a good question to consider the case when the spectrum of the Fourier transform of a compact supported function is supported on an uncountably thin set. That is, the support of an integrable function f and its Fourier transform ˆf cannot both be of finite measure simultaneously. The Fourier transform on a noncommutative set becomes a linear operator of a large rank in contrast to Euclidean spacesRn where the Fourier transform has rank one and thus can be thought of as a function on Rn.
In this thesis, we prove that if the group Fourier transform of certain integrable functions on the Euclidean motion groups (or the Heisenberg motion group/stage two nilpotent Lie groups) is of finite rank, then the function must be identically zero. These results can be thought of as an analogue of Benedick's theorem, which was about the uniqueness of the Fourier transform of integrable functions on Euclidean spaces.
