It is a well known fact that the generalized Vandermonde determinant can be expressed as the product of the standard Vandermonde determinant and a polynomial with nonnegative integer coefficients. In this paper we generalize this result to Vandermonde determinants over the Chebyshev basis. We apply this result to prove that the number of real roots in [1;1] of a real polynomial is bounded by th...

Abstract Finite elements with polynomial basis functions on the simplex a symmetric distribution of nodes should have unique representation. Unisolvence not only requires that number equals independent polynomials spanning space given degree, but also Vandermonde matrix controlling their mapping to Lagrange interpolating can be inverted. Here, necessary condition for unisolvence is presented sp...

This paper contains two fast algorithms for inversion of ChebyshevvVander-monde matrices of the rst and second kind. They are based on special representations of the Bezoutians of Chebyshev polynomials of both kinds. The paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster, but also more stable than other algorithms avail...

In this correspondence we answer a question posed in [WT99] and provide an elementary proof of the Kharitonov theorem deducing it from the classical Hermite criterion. The proof is based on the concept of a Bezoutian matrix. Generally, exploiting the special structure of such matrices (e.g., Bezoutians, Toeplitz, Hankel or Vandermonde matrices, etc.) can be interesting, e.g., leading to unified...

Polynomial based methods have recently been used in several works for mitigating the effect of stragglers (slow or failed nodes) distributed matrix computations. For a system with $n$ worker nodes where notation="LaTeX">$s$ can be stragglers, these approaches allow an optimal recovery threshold, whereby intended result dec...

The purpose of this paper is to study the conditioning of complex Vandermonde matrices, in reference to applications such as superresolution and the problem of recovering missing samples in band-limited signals. The results include bounds for the singular values of Vandermonde matrices whose nodes are complex numbers on the unit circle. It is shown that, under certain conditions, such matrices ...

Recently the one-dimensional time-discrete blind deconvolution problem was shown to be solvable uniquely, up to a global phase, by a semi-definite program for almost any signal, provided its autocorrelation is known. We will show in this work that under a sufficient zero separation of the corresponding signal in the z−domain, a stable reconstruction against additive noise is possible. Moreover,...

The Galois ring GR(p,m) of characteristic p and cardinality p, where p is a prime and r,m ≥ 1 are integers, is a Galois extension of the residue class ring Zpr by a root ω of a monic basic irreducible polynomial of degree m over Zpr . Every element of GR(p ,m) can be expressed uniquely as a polynomial in ω with coefficients in Zpr and degree less than or equal to m − 1, thus GR(p,m) is a free m...

It is known that the space of real valued, continuous functions C(B) over a multidimensional compact domain B ⊂ R , k ≥ 2 does not admit Haar spaces, which means that interpolation problems in finite dimensional subspaces V of C(B) may not have a solutions in C(B). The corresponding standard short and elegant proof does not apply to complex valued functions over B ⊂ C. Nevertheless, in this sit...