Fourier transform, null variety, and Laplacian's eigenvalues

Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals

It is known that Laplacian operators on many fractals have gaps in their spectra. This fact precludes the possibility that a Weyl-type ratio can have a limit and is also a key ingredient in proving that the Fourier series on such fractals can have better convergence results than in the classical setting. In this paper we prove that the existence of gaps is equivalent to the total disconnectedness of the Julia set of the spectral decimation function for the class of fully symmetric p.c.f. fractals, and for self-similar fully symmetric finitely ramified fractals with regular harmonic structure. …

由图短时傅里叶变换生成的紧框架

Tight Frames Generated By A Graph Short-Time Fourier Transform

本文利用图拉普拉斯特征向量和图热核定义图短时傅里叶变换，证明其构成框架并刻画框架算子谱，还证明对两类代数图该框架是紧的且与时间参数无关。

Dirichlet Laplacian 的 Weyl 定律

The Weyl law for the Dirichlet Laplacian

本文回顾 Dirichlet Laplacian 特征值的渐近分布，引入相关谱量，并基于 Fourier Tauberian 定理给出 Weyl 定律的证明。

Dual of 2D fractional Fourier transform associated to Itô--Hermite polynomials

A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-S…

How can we naturally order and organize graph Laplacian eigenvectors?

When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequencies of the corresponding eigenvectors. In this paper, we discuss this important problem further and propose a new method to organize those eigenvectors by defining and measuring "natural" distances betwe…

The semiclassical structure of low-energy states in the presence of a magnetic field

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier components of an operator with the structure of the Szegö projector, i.e. a Fourier integral operator of Hermite type. This result yields semiclassical asymptotics for the low-energy eigenstates.