Related papers: Radial positive definite functions and Schoenberg …

We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

For positive integers $1 \leq k \leq n$ let $M_n$ be the algebra of all $n \times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most $k$. We first show that whenever $k>\frac{n}{2}$, any continuous…

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha…

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional…

I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding…

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

We define an abelian group homomorphism $\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\mathscr{F}$ in the Schur basis are the…

We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg,…

We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain $\Omega$ of a rectangular domain $I\times I$ to a covariance kernel on the entire domain $I\times I$. For a…

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$.…

Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s)<s\ \text{for}\ s\in (0,\beta), \ f(s)>s\ \text{for}\ s\in (\beta, \infty)$ and…

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not…

Given an infinite sequence of positive integers $\cA$, we prove that for every nonnegative integer $k$ the number of solutions of the equation $n=a_1+...+a_k$, $a_1,\,..., a_k\in \cA$, is not constant for $n$ large enough. This result is a…

The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its…

This article is concerned with the following spectral problem: to find a positive function $\Phi$ $\in$ C 1 ($\Omega$) and $\lambda$ $\in$ R such that q(x)$\Phi$ (x) + ^ $\Omega$ J(x, y)$\Phi$(y) dy + a(x)$\Phi$(x) + $\lambda$$\Phi$(x) = 0…

Assume that $\Delta_h$ is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ and assume that $\Phi_n$ is the unique radial solution of Poisson equation $\Delta_h \log \Phi_n =-4 (n-1)^2$ satisfying the condition $\Phi_n(0)=1$ and…

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite…