arXiv subjects
Publications and source records attributed to Emmett Wyman.
We investigate whether or not one can hear at what point (up to symmetry) a drum is struck for some special planar triangles. We prove that one can hear where the equilateral and isosceles right triangles are struck. We also prove that the 30-60-90 triangle, surprisingly, has two audibly indistinguishable points. This is the first known topologically connected such example.
The purpose of this paper is to develop a Fourier uncertainty principle on compact Riemannian manifolds and contrast the underlying ideas with those arising in the setting of locally compact abelian groups. The key obstacle is the growth of eigenfunctions, and connections to Bourgain's celebrated $Λ_q$ theorem are discussed in this context.
This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ϕ_j$ of a compact Riemannian manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and study the asymptotics of the thin sums, $$ N^{c} _{ε, H }(λ): = \sum_{j, λ_j \leq λ} \sum_{k: |μ_k - c λ_j | < ε} \left| \int_{H} ϕ_j \overline{ψ_k}dV_H \right|^2 $$ where $\{λ_j\}$ are the eigenvalues of $\sqrt{-Δ}_M,$ and $\{(μ_k, ψ_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-Δ}_H$. The inner sums represent the `jumps' of $ N^{c} _{ε, H }(λ)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $ξ\in S^c_H M, π_H ξ\in B^* H$, where $π_H ξ$ is the orthogonal projection of $ξ$ to $H$. A c-bi-angle occurs when $\frac{|π_H ξ|}{|ξ|} = c$. Smoothed sums in $μ_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $ε$ varies, at certain values of $ε$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k: | \frac{μ_k}{λ_j} - c| \leq ε$ and where geodesic bi-angles do not play a role.
The classical Weyl Law says that if $N_M(λ)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $λ$, then $$ N_M(λ)=cλ^d+O(λ^{d-1}).$$ In this paper, we show Duistermaat and Guillemin's result allows us to replace the $O(λ^{d-1})$ error with $o(λ^{d-1})$ if $M$ is a product manifold. We quantify this bound in the case of Cartesian product of spheres by reducing the problem to the study of the distribution of weighted integer lattice points in Euclidean space and formulate a conjecture in the general case reminiscent of the sum-product phenomenon in additive combinatorics.
In this paper we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the first example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups.