Related papers: Matrix Fej\'er-Riesz type theorem for a union of a…
The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ by the use of matrix…
We prove a Fej\'er-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product…
Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejer-Riesz…
A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case
Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published. These theorems characterize the symbols that give rise to finite rank multidimensional Hankel and Toeplitz type operators…
In this paper, we present a constructive proof of Popescu's non-commutative Fej\'er-Riesz theorem for non-commuting polynomials. We are considering non-commutating polynomial in left-creation and left-annihilation multi-Toeplitz operators.
In this paper, several versions of the Kolmogorov-Riesz compactness theorem in weighted Lebesgue spaces with matrix weights are obtained. In particular, when the matrix weight $W$ is in the known $A_p$ class, a characterization of totally…
Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear…
We present an expanded expository account of the $K$-moment problem for polynomial algebras over \(\R^d\), with special emphasis on compact basic closed semialgebraic sets. The central question is to characterize those linear functionals on…
In the classical Hardy space theory of square-summable Taylor series in the complex unit disk there is a circle of ideas connecting Szeg\"o's theorem, factorization of positive semi-definite Toeplitz operators, non-extreme points of the…
This paper studies Positivstellens\"atze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$…
For an infinite Toeplitz matrix $T$ with nonnegative real entries we find the conditions, under which the equation $\boldsymbol{x}=T\boldsymbol{x}$, where $\boldsymbol{x}$ is an infinite vector-column, has a nontrivial bounded positive…
In this paper, we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, if they exist. We show that when the…
We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters,…
Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem…
The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure…
We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…
A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported?…
We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the…
Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all…