Related papers: Decrease of Fourier coefficients of stationary mea…

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset…

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the…

In this note, we generalise a Bourgain's construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of $\mathrm{SL}_2(\mathbb{R})$ to that of a general simple Lie group. The proof is…

Let $\Gamma$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $\Gamma$-space and $\mu$ and $\nu$ are Borel probability measures on $X$ which agree on every $\Gamma$-invariant subset, then $\mu$…

We consider partial theta series associated with periodic sequences of coefficients, of the form $\Theta(\tau) := \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}$, with $\nu$ non-negative integer and an $M$-periodic function $f : \mathbb{Z}…

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for…

Let $\mu$ be a given Borel measure on $\K\subseteq\R^n$ and let $y=(y_\alpha)$, $\alpha\in\N^n$, be a given sequence. We provide several conditions linking $y$ and the moment sequence $z=(z_\alpha)$ of $\mu$, for $y$ to be the moment…

Let $(\Omega,\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset\Omega\times\Omega$ be a measurable equivalence relation, that is,…

Given any function $\phi \colon [0,\infty)\to (0,1]$ satisfying $\lim_{\xi\to\infty}\phi(\xi) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose…

Given a second order parabolic operator $$ Lu(t,x) :=\frac{\partial u(t,x)}{\partial t} + a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x) + b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}\mu=0$ for Borel…

We study the probability measure $\mu_{0}$ for which the moment sequence is $\binom{3n}{n}\frac{1}{n+1}$. We prove that $\mu_{0}$ is absolutely continuous, find the density function and prove that $\mu_{0}$ is infinitely divisible with…

We continue the study (initiated in [1]) of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficency in the measure caused by…

We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $\nu$ on $K$ minimizing $\int_K f d\nu$.…

We generalize the Beckner's type Poincar\'e inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $\mu\star\nu$, where $\mu$ is the reference Gaussian measure and $\nu$ is a probability…

This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The…

We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages…

Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval $J$ and two sequences $(n_k)_{k=1}^\infty$ and $(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite product measure (Moran measure) on $E$…

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear…

In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We…