Related papers: Hirschman-Widder densities

Let $G$ be an LCA group, $H$ a closed subgroup, $\varGamma$ the dual group of $G$ and $\mu$ be a regular finite non-negative Borel measure on $\varGamma$. We give some necessary and sufficient conditions for the density of the set of…

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…

Let $\mathbf a=(a_1,\ldots,a_r)$ be a sequence of positive integers and $k\geq 2$ an integer. We study $p_{k,\mathbf a}(n)$, the restricted $k$-multipartition function associated to $\mathbf a$ and $k$. We prove new formulas for…

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and H\"{o}lder continuous drift has a H\"{o}lder continuous density function. This result complements recent…

In previous works we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness…

A Fourier-type integral representation for Bessel's function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson's integral representation for the Bessel function along with a trigonometric integral…

In combinatorics, P\'{o}lya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of P\'{o}lya's…

We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this…

The core focus of this series of two articles is the study of the distribution of the length spectrum of closed hyperbolic surfaces of genus $g$, sampled randomly with respect to the Weil-Petersson probability measure. In the first article,…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(\mu)$ for any $\mu$ such that the moments $\int x^k d\mu$ do not grow too rapidly as $k \to \infty$. In this work, we…

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…

We prove an extension of Hoermander's classical result on hypoelliptic second order equations, where the coefficients of the related vector fields are globally Lipschitz and satisfy the classical Hoermander condition on a dense set while…

Special bases of orthogonal polynomials are defined, that are suited to expansions of density and potential perturbations under strict particle number conservation. Particle-hole expansions of the density response to an arbitrary…

Let $\lambda(n)$ be the Liouville function. We study the distribution of \[ \frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n)) \] over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular…

We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation…

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The…

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of…

Given ideals $I,J$ of a noetherian local ring $(R, \mathfrak m)$ such that $I+J$ is $\mathfrak m$-primary and a finitely generated $R$-module $M$, we associate an invariant of $(M,R,I,J)$ called the $h$-function. Our results on…

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we…