in this paper we demonstrate the existence of a set of polynomials pi , 1 i  n , which arepositive semi-definite on an interval [a , b] and satisfy, partially, the conditions of polynomials found in thelagrange interpolation process. in other words, if a  a1  an  b is a given finite sequence of realnumbers, then pi (a j )  ij (ij is the kronecker delta symbol ) ; moreover, the sum of ...

A univariate trace polynomial is a in variable x and formal symbols Tr ( j ) . Such an expression can be naturally evaluated on matrices, where the are as normalized traces. This paper addresses global constrained positivity of polynomials symmetric matrices all finite sizes. tracial analog Artin's solution to Hilbert's 17th problem given: positive semidefinite quotient sums products squares tr...

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...

Associated to a squarematrix all of whose entries are real Laurent polynomials in several variables with no negative coefficients is an ordered “dimension” module introduced by Tuncel, with additional structure, which acts as an invariant for topological Markov chains, and is also an invariant for actions of tori on AF C*-algebras. In describing this invariant, we are led naturally to eventuall...

There are various reasons for the interest in the problem of constructing nonnegative trigonometric polynomials. Among them are: Cesàro means and Gibbs’ phenomenon of the the Fourier series, approximation theory, univalent functions and polynomials, positive Jacobi polynomial sums, orthogonal polynomials on the unit circle, zero-free regions for the Riemann zeta-function, just to mention a few....

All polynomials in this paper are supposed to have real coefficients. Polynomials which can be represented in the form P(X) = c ad1-x) " U + XY, with all akl > 0 or all akl < 0, (1) k+l 0 have been introduced and studied by 6.6. Lorentz i[ I]; we shall call them polynomials with positive or negative (more exactly non-negative or non-positive) coefficients, respectively, or simply Lorent...