无穷矩阵的迭代函数系统、矩和变换

迭代函数系测度的拟周期谱与正交性

Quasiperiodic Spectra and Orthogonality for Iterated Function System Measures

我们将经典的基础结构从傅里叶分析推广到仿射迭代函数系统（IFS）的吸引子。这是有趣的，因为这些吸引子具有分形特征，例如，用分形标度维数来度量。此外，频谱通常是准周期性的，但是非周期性的，即，光谱是晶格的"小扰动“。由于对IFS的早期研究，在某些类别的谱对偶对（也称为谱对或谱测度）上有已知的结果。已知某些对偶对与复Hadamard矩阵有关。然而，并非所有IFS $X$都承认谱对偶性。当$X$给出，我们确定的几何条件$X$的傅立叶谱的存在，作为第二部分的频谱对。我们展示了这些光谱对如何组成，我们的特点分解的原子。分解是指相关复Hadamard矩阵的张量积因子分解。

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(α)$ is non-negative if and only if $α$ is the moment sequence of a measure $μ$ on $\mathbb{R}^\infty$, assuming that $α$ does not grow too fast. We then characterize the non-negative bounded H…

Inverses of moment Hermitian matrices

Motivated by [9] we study the existence of the inverse of infinite Hermitian moment matrices associated with measures with support on the complex plane. We relate this problem to the asymptotic behaviour of the smallest eigenvalues of finite sections and we study it from the point of view of infinite transition matrices associated to the orthogonal polynomials. For Toeplitz matrices we introduce the notion of weakly asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of a Toeplitz moment matrix is weakly asymptotic Toeplitz. Such inverses are computed in terms o…

非负矩阵乘积的压力函数

The Pressure Function for Products of Non-negative Matrices

设$(Σ_A, σ)$是有限型子移位，$M(x)$是$Σ_A$上取非负矩阵集合中值的连续函数.我们将经典的标量压力函数推广到这个新的情形，证明了吉布斯测度的存在性和压力函数的可微性。我们特别感兴趣的情况下，$M(x)$采取有限值$M_1, ..., M_m$。压力功能降至$P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$。当我们考虑某些具有重叠的迭代函数系统的多重分形形式时，该表达式是重要的。

Optimal transportation and stationary measures for Iterated Function Systems

In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a "…

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one …