Related papers: Generalized Bessel and Frame Measures

A metric measure space $(X,d,\mu)$ is said to be $A_{\infty}$ on curves if there exist constants $\tau < 1$ and $\theta > 0$ with the following property. For every $x \in X$, $0 < r \leq \mathrm{diam}(X)$, and a Borel set $S \subset B(x,r)$…

For a class of one-dimensional determinantal point processes including those induced by orthogonal projections with integrable kernels satisfying a growth condition, it is proved that their conditional measures, with respect to the…

The paper deals with various centering problems for probability measures on finite dimensional vector spaces. We show that for every such measure there exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each symmetry $S$…

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class…

The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…

Let $\alpha>0$ and $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\ge0}$ with entries…

Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and let $d\mu$ be a probability measure with…

We provide an algorithm to approximate a finitely supported discrete measure $\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence if…

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\gamma>0$, the Hankel matrix $\mathcal{H}_{\mu,\gamma}=(\mu_{n,k})_{n,k\geq0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n+k}=\int_{0}^{\infty}t^{n+k}d\mu(t)$.…

Let $\mu$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $\mu $. We establish…

Let $L_{q,\mu}$, $1\leq q\leq\infty$, denotes the weighted $L_q$ space of functions on the unit ball $\Bbb B^d$ with respect to weight $(1-\|x\|_2^2)^{\mu-\frac12},\,\mu\ge 0$, and let $W_{2,\mu}^r$ be the weighted Sobolev space on $\Bbb…

For any pair of bounded observables $A$ and $B$ with pure point spectra, we construct an associated "joint observable" which gives rise to a notion of a joint (projective) measurement of $A$ and $B$, and which conforms to the intuition that…

We introduce the categories of quasi-measurable spaces, which are slight generalizations of the category of quasi-Borel spaces, where we now allow for general sample spaces and less restrictive random variables, spaces and maps. We show…

Let $\mu$ be a positive measure on $R^d$. It is known that if the space $L^2(\mu)$ has a frame of exponentials then the measure $\mu$ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. It has been…

For all $s\in[0,1]$ and $t\in(0,s]\cup [2-s,2)$, we find the supremum of numbers $\omega\in(0,2)$ such that $\text{I}_\omega(\mu\ast\sigma) \lesssim 1$, where $\mu$ is any Borel measure on $B(1)$ with $\text{I}_t(\mu)\leq 1$ and $\sigma$ is…

For any standard Borel space $B$, let $\mathcal{P}(B)$ denote the space of Borel probability measures on $B$. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin…

We prove the following isoperimetric type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb R}^n$, halfspaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress…

We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…

In this paper, we consider the weighted Hardy space $\mathcal{H}^p(\omega)$ induced by an $A_1$ weight $\omega.$ We characterize the positive Borel measure $\mu$ such that the identical operator maps $\mathcal{H}^p(\omega)$ into $L^q(d\mu)$…

We give a complete characterization of the size of Borel sets that are mid-point convex but not (essentially) convex, in terms of their Hausdorff dimensions and Hausdorff measures.