相关论文: On spectral Cantor measures
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by…
This paper investigates the problem of estimating the spectral power parameters of random analog sources using numerical measurements acquired with minimum digitization complexity. Therefore, spectral analysis has to be performed with…
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2…
In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category.…
Given any two probability measures on a Euclidean space with mean 0 and finite variance, we demonstrate that the two probability measures are orthogonal in the sense of Wasserstein geometry if and only if the two spaces by spanned by the…
Homemade spectrometers are commonly used tools to analyze light sources and determine its physical characteristics. We perform an assessment of homemade spectrometers in terms of spectral resolution and accuracy in the determination of…
Complete spectroscopy (measurements of a complete sequence of consecutive levels) is often considered as a prerequisite to extract fluctuation properties of spectra. It is shown how this goal can be achieved even if only a fraction of…
We consider the spectral analysis of several examples of bilateral birth-death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some…
Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis…
In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent…
We introduce and study a new theoretical concept of \textit{spectral pair} for a Schr\"{o}dinger operator $H$ in $L^2(\mathbb{R}_{+})$ with a bounded \textit{complex-valued} potential. The spectral pair consists of a scalar measure and a…
The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are…
We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for…
We study spectral properties of the self-affine measure $\mu_{M,\mathcal {D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in…
We study two spiked models of random matrices under general frameworks corresponding respectively to additive deformation of random symmetric matrices and multiplicative perturbation of random covariance matrices. In both cases, the…
We suggest scattering experiments which implement the concept of ``protective measurements'' allowing the measurement of the complete wave function even when only one quantum system (rather than an ensemble) is available. Such scattering…
Let $\mu$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure type. We…
Let $\left\{b_{k}\right\}_{k=1}^{\infty}$ be a sequence of integers with $|b_{k}|\geq2$ and $\left\{D_{k}\right\}_{k=1}^{\infty} $ be a sequence of equidifferent digit sets with $D_{k}=\left\{0,1, \cdots, N-1\right\}t_{k},$ where $N\geq2$…
We develop the basic theory of ergodic Schr\"odinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and…