Related papers: Extension of positive definite functions

We study maximal representations of nonnegative sesquilinear forms in real or complex Hilbert spaces, that are not necessarily closed or even closable. We associate positive self-adjoint operators with such forms, in a sense similar to…

Let $\Omega^o$ and $\Omega^i$ be open bounded subsets of $\mathbb{R}^n$ of class $C^{1,\alpha}$ such that the closure of $\Omega^i$ is contained in $\Omega^o$. Let $f^o$ be a function in $C^{1,\alpha}(\partial\Omega^o)$ and let $F$ and $G$…

A positive definiteness criterion and, under the additional conditions, a nonnegativity criterion for a self-adjoint continuous operator matrix, acting in product of an arbitrary number of real separable Hilbert spaces, are obtained. As…

Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be…

Suppose we are given complex manifolds $X$ and $Y$ together with substacks $\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation $\mathcal{A}$ on $X$ and $\mathcal{A}'$ on $Y$, respectively. Suppose also we are…

We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the…

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth…

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for…

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin…

We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the…

Arakeljan's Theorem provides conditions on a relatively closed subset $F$ of a domain $G\subset\mathbb{C}$, such that any continuous function $f:F\rightarrow\mathbb{C}$ that is analytic in $F^\circ$, can be approximated by analytic…

Let $U_{n}\subset C^{n}[ a,b] $ be an extended Chebyshev space of dimension $n+1$. Suppose that $f_{0}\in U_{n}$ is strictly positive and $% f_{1}\in U_{n}$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for…

This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum…

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

We formulate and prove in this report some sufficient conditions for exponential tightness (ET) of a family of independent identical distributed (i.i.d.) random fields (r.f.) (processes) in the space of continuous functions defined on…

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is…

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…

The stretched Gau{\ss}ian function $f(\mathbf{x})=\exp \left(-\|\mathbf{x}\|^s\right)$, as a real function defined on $\mathbb{R}^d$, has found numerous applications in mathematics and physics. For instance, to describe results from…

We give extensional and intensional characterizations of functional programs with nondeterminism: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which…