Borel Measure Research Articles

Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Let $$\{S_j: 1\le j\le 3\}$$ be a set of three contractive similarity mappings such that $$S_j(x)=rx+\frac{j-1}{2}(1-r)$$ for all $$x\in {\mathbb {R}}$$ , and $$1\le j\le 3$$ , where $$0<r<\frac{1}{3}$$ . Let $$P=\sum _{j=1}^3 \frac{1}{3} P\circ S_j^{-1}$$ . Then, P is a unique Borel probability measure on $${\mathbb {R}}$$ such that P has support the Cantor set generated by the similarity mappings $$S_j$$ for $$1\le j\le 3$$ . Let $$r_0=0.1622776602$$ , and $$r_1=0.2317626315$$ (which are ten digit rational approximations of two real numbers). In this paper, for $$0<r\le r_0$$ , we give a general formula to determine the optimal sets of n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers $$n\ge 2$$ . Previously, Roychowdhury gave an exact formula to determine the optimal sets of n-means and the nth quantization errors for the standard triadic Cantor distribution, i.e., when $$r=\frac{1}{5}$$ . In this paper, we further show that $$r=r_0$$ is the greatest lower bound, and $$r=r_1$$ is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for $$0<r\le r_1$$ the quantization coefficient does not exist though the quantization dimension exists.

Polar forcings and measured extensions

An ideal I on a Polish space X is said to be polar if I is the intersection of the null ideals for some family of Borel probability measures on X. We study polar ideals where the corresponding family of measures is analytic and the induced forcing of Borel sets modulo I is proper. We show that for a broad class of examples this property is closed under iterations, and that the universally measurable sets of the ground model reinterpret as universally measurable sets in the corresponding extensions.

Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

AbstractA bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

Least Wasserstein distance between disjoint shapes with perimeter regularization

We prove the existence of global minimizers to the double minimization problem [Display omitted] where P(E) denotes the perimeter of the set E, Wp is the p-Wasserstein distance between Borel probability measures, and λ>0 is arbitrary. The result holds in all space dimensions, for all p∈[1,∞), and for all positive λ. This answers a question of Buttazzo, Carlier, and Laborde.

The Period Group of a Characteristic Function

Let $\widehat{\mu}$ be the Fourier transform of a Borel probability measure $\mu$ on $\mathbb R^d$. We look at the closed abelian subgroup $\Gamma (\mu)$ of $\mathbb R^d$, which consists of the periods of the function $\widehat{\mu}$. We prove the following dichotomy: i) The support of $\mu$ is non-degenerate if and only if $\Gamma (\mu)$ is a lattice. ii) The support of $\mu$ is degenerate if and only if $\Gamma (\mu)$ contains a linear subspace $\neq \{0\}$ of $\mathbb R^d$. A similar dichotomy is also discussed for the period group of the function $|\widehat{\mu}|$.

Compact spaces associated to separable Banach lattices

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what C(K) spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum K admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of beta {{mathbb{N}}}. Some natural questions about this class are left open.

Packing Entropy of Saturated Sets for Nonuniformly Hyperbolic Systems

In this paper, we prove that for a nonuniformly hyperbolic system [Formula: see text], and for every convex and compact subset [Formula: see text] with the same hyperbolic rate, the packing entropy of saturated set [Formula: see text] equals the supremum of the metric entropy of all measures in K. Here [Formula: see text] denotes the set of all [Formula: see text]-invariant Borel probability measures supported on [Formula: see text]. As an application, we describe the size of the set of mean Li–Yorke pairs.

Viability and invariance of systems on metric spaces

We consider a generalized control system on a metric space and investigate necessary and sufficient conditions for viability and invariance of proper subsets, describing state constraints. Viability means that for every initial condition in the set of constraints we can find trajectories of control system starting at this condition and satisfying state constraints forever. Invariance means that every trajectory of control system starting in the set of constraints never violates them. As examples of application we consider controlled continuity equations on the metric space of Borel probability measures having compact support, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compacts subsets of the Euclidean space endowed with the Hausdorff metric. We also provide sufficient conditions for the existence and uniqueness of contingent solutions to the Hamilton–Jacobi–Bellman equation on proper metric spaces.

ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES A_{mu }^1

We consider weighted Bergman spaces A_mu ^1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of A_mu ^1. Also, as a consequence of a characterization of solid A_mu ^1-spaces, we show that, in the case of entire functions, there indeed exist solid A_mu ^1-spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of A_mu ^1, when mu equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A^1- and H^infty-spaces, the validity of which requires the assumption that -log v belongs to the class mathcal {W}_0, studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight v(z) = exp ( -1 /(1-|z|) provides an example satisfying both assumptions.

Stability theorems in pointwise dynamics

We introduce minimally expansive and GH-stable points for homeomorphisms on metric spaces and μ-uniformly expansive, μ-shadowable and strong μ-topologically stable points for Borel measures (with respect to a homeomorphism on a metric space). We prove that: (i) a minimally expansive shadowable point of a homeomorphism on a compact metric space is topologically stable and GH-stable; (ii) a μ-uniformly expansive μ-shadowable point for a Borel measure μ (with respect to a homeomorphism on a compact metric space) is strong μ-topologically stable.

Topological properties in tensor products of Banach spaces

Given two Banach spaces X and Y, we analyze when the projective tensor product X⊗ˆπY has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X⊗ˆπY is WLD if and only if both X and Y are WLD and all operators from X to Y⁎ and from Y to X⁎ have separable range; (ii) X⊗ˆπY is subspace of a WCG space if the same holds for both X and Y under the assumption that every operator from X to Y⁎ is compact; (iii) ℓp(Γ)⊗ˆπℓq(Γ) is subspace of a Hilbert generated space for any 1<p,q<∞ such that 1/p+1/q<1 and for any infinite set Γ. We also pay attention to the injective tensor product X⊗ˆεY. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if K is a compact space such that C(K×K) has property (C), then all regular Borel probability measures on K have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the w⁎-angelicity of the dual unit ball under injective tensor products.

Large sets without Fourier restriction theorems

We construct a function that lies in L p ( R d ) L^p(\mathbb {R}^d) for every p ∈ ( 1 , ∞ ] p \in (1,\infty ] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.

Integral comparisons of nonnegative positive definite functions on LCA groups

In this paper we investigate the following questions. Let mu , nu be two regular Borel measures of finite total variation. When do we have a constant C satisfying ∫fdν≤C∫fdμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int f d\ u \\le C \\int f d\\mu \\end{aligned}$$\\end{document}whenever f is a continuous nonnegative positive definite function? How the admissible constants C can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures mu , nu are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove a duality conjecture posed in our former paper.

Heavy traffic scaling limits for shortest remaining processing time queues with heavy tailed processing time distributions

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. Since one needs to keep track of the remaining processing times of all jobs in the system in order to describe the evolution, a natural state descriptor for an SRPT queue is a measure valued process in which the state of the system at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system. In this work we are interested in studying the asymptotic behavior of the suitably scaled measure valued state descriptors for a sequence of SRPT queuing systems. Gromoll, Kruk and Puha (Stoch. Syst. 1 (2011) 1–16) have studied this problem under diffusive scaling (time is scaled by r2 and the mass of the measure normalized by r, where r is a scaling parameter approaching infinity). In the setting where the processing time distributions have bounded support, under suitable conditions, they show that the measure valued state descriptors converge in distribution to the process that at any given time is a single atom located at the right edge of the support of the processing time distribution with the size of the atom fluctuating randomly in time. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measure valued state descriptors converge in distribution to the process that is identically zero. In Puha (Ann. Appl. Probab. 25 (2015) 3381–3404) for the setting where the processing time distributions have unbounded support and light tails, a nonstandard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit. In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. Results of Puha (Ann. Appl. Probab. 25 (2015) 3381–3404) suggest that the right scaling for the measure valued process is governed by a parameter cr that is given as a certain inverse function related to the tails of the first moment of the processing time distribution. Using this parameter we consider a novel scaling for the measure valued process in which the time is scaled by a factor of r2, the mass is scaled by the factor cr/r and the space (representing the remaining processing times) is scaled by the factor 1/cr. We show that the scaled measure valued process converges in distribution (in the space of paths of measures). In a sharp contrast to results for bounded support and light tailed service time distributions, this time there is no state space collapse and the limiting measures are not concentrated on a single atom. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain R+ valued random field which is determined from a single Brownian motion. Along the way we establish convergence of suitably scaled workload and queue length processes. We also show that as the tail of the distribution of job processing times becomes lighter in an appropriate fashion, the difference between the limiting queue length process and the limiting workload process converges to zero, thereby approaching the behavior of state space collapse.

Extremal functions and invariant subspaces in Dirichlet spaces

Let μ be a positive finite Borel measure on the unit circle of the complex plane, and let D(μ) be the associated Dirichlet space. The Beurling-Deny capacity associated with D(μ) is denoted by cμ. Brown-Shields conjecture for D(μ) says that a function f∈D(μ) is cyclic for D(μ), meaning that {pf:pis a polynomial} is dense in D(μ), if and only if f is outer and the boundary zeros set of f is of cμ− capacity zero. It was recently proved, using Banach algebras techniques, that Brown-Shields conjecture is true for D(μ) whenever the support of μ is countable. In this paper, we produce a new class of measures for which Brown-Shields conjecture holds. This class includes the canonical measures on Cantor sets. For these measures, we also give an explicit description of all invariant subspaces for the shift operator, i.e. the operator of multiplication by the independent variable. Our method is based on a study of the behavior of extremal functions for Dirichlet spaces. We prove that if the support of μ is a Carleson set, then every outer extremal function φ can be explicitly recovered from the restriction of its boundary values |φ⁎| on the support of μ. This allows us to give estimates of φ for a large class of measures.

The strong truncated Hamburger moment problem with and without gaps

The strong truncated Hamburger moment problem (STHMP) of degree (−2k1,2k2) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R, such that βi=∫xidμ(−2k1≤i≤2k2). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. Then, using the equivalence with the STHMP of degree (−2k,2k), we obtain the solution of the 2–dimensional truncated moment problem (TMP) of degree 2k with variety xy=1, first solved by Curto and Fialkow [22]. Our addition to their result is the fact previously known only for k=2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k1,2k2) with one missing moment in the sequence, i.e., β−2k1+1 or β2k2−1, which also gives the solution of the TMP with variety x2y=1 as a special case, first studied by Fialkow in [33].

Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals

Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G, which is also a dense linear subspace of H, and then by taking its dual G*, we obtain a real Gel’fand triple G⊂H⊂G*. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γN on G* such that its covariance operator coincides with N. We examine the properties of γN, and, among others, we show that γN can be represented as a convolution of a sequence of Borel probability measures on G*. Some other results are also obtained.

Isobarycentric Inequalities

Abstract We prove the following isoperimetric-type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb {R}}^n$, half-spaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.

The Borel measure of sequences with bounded run-length

The Borel measure of sequences with bounded run-length

Exceptional sets for average conformal dynamical systems

Let f : M → M be a C1+α local diffeomorphism/diffeomorphism of a compact Riemannian manifold M and μ be an expanding/hyperbolic ergodic f-invariant Borel probability measure on M. Assume f is average conformal expanding/hyperbolic on the support set W of μ and W is locally maximal. For any subset A ⊂ W with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the A-exceptional set and the limit A-exceptional set.