Related papers: The moment problem with bounded density

Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \Psi(t) be a positive continuous function such that \|\Psi f^{\beta}\|_{\infty}<\infty for some 0<\beta<1/2. Under natural…

Let $(\mathfrak{M},\rho,\mu)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},\mu)\rightarrow L^{p_0}(\mathfrak{M},\mu)$, $t\geq 0$, a strongly continuous semi-group. We provide…

We enquire under which conditions, given two $\sigma$-finite, $\omega$-continuous valuations $\nu$ and $\mu$, $\nu$ has density with respect to $\mu$. The answer is that $\nu$ has to be absolutely continuous with respect to $\mu$, plus a…

We show that under very mild conditions on a measure $\mu$ on the interval $[0,\infty)$, the span of $\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ for any $n=0,1,\ldots$. We present two different proofs of this result, one based on the…

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$…

In this paper we study Devinatz's moment problem: to find a non-negative Borel measure $\mu$ in a strip $\Pi = \{(x,\phi):\ x\in \mathbb{R},\ -\pi\leq \phi < \pi\},$ such that $\int_\Pi x^m e^{in\phi} d\mu = s_{m,n}$, $m\in \mathbb{Z}_+$,…

A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $\rho$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k \rho(1-|z_k|)|f(z_k)|$ converges. All sequences…

A permutation sequence $(\sigma_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $\tau$, the density of occurrences of $\tau$ in the elements of the sequence converges. We prove that such a convergent sequence…

For $\ell\colon \mathbb{R}^d \to [0,\infty)$ we consider the sequence of probability measures $\left(\mu_n\right)_{n \in \mathbb{N}}$, where $\mu_n$ is determined by a density that is proportional to $\exp(-n\ell)$. We allow for infinitely…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either…

Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\mathcal{B}, \mu)$ such that $\sum_{n=1}^{\infty} \mu (B_{i}) = \infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\mu…

We prove that every finite Borel measure $\mu$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $\mu\lfloor_{E_k}$ that are bounded from above by the Hausdorff…

A quantum system consisting of two subsystems is separable if its density matrix can be written as $\rho=\sum w_K \rho_K'\otimes \rho_K''$, where $\rho_K'$ and $\rho_K''$ are density matrices for the two subsytems, and the positive weights…

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(\Omega,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then…

We study a class of positive random variables having moments of Gamma type, whose density can be expressed by the three-parametric Mittag-Leffler functions. We give some necessary conditions and some sufficient conditions for their…

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence…

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…