In this paper, we review some potential applications of random Vandermonde matrices in the field of signal processing and wireless communications. Using asymptotic results based on the theory of random Vandermonde matrices, we show through several application examples, namely deconvolution, wireless capacity analysis and sampling theory, the research potential of this theory. Quite surprisingly...

Multipoint polynomial evaluation and interpolation are fundamental for modern numerical and symbolic computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. We decrease this cost dramatically and for a large class of inputs yield nearly linear time as well. We first restate our task...

p/k + 4b + c. Our proof is inspired by the so called Stepanov method, which involves bounding the size of the solution set of a system of equations by constructing a nonzero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number theory along with a determinant identity that generalizes both the Sylvester and the Vandermo...

We combine Young idempotents in the group algebra of the symmetric group with the action of the symmetric group on products of Vandermonde determinants to obtain idempotents with polynomial coefficients.

In this paper, we determine some limit distributions of pattern statistics in rational stochastic models. We present a general approach to analyze these statistics in rational models having an arbitrary number of strongly connected components. We explicitly establish the limit distributions in most significant cases; they are characterized by a family of unimodal density functions defined by me...

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices can date back to the early twentieth century. It forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vander...

that is as easy as playing cards. Let Vn denote the Vandermonde matrix with (i, j)th entry vi j = x j i (0 ≤ i, j ≤ n). Since the determinant of Vn is a polynomial in x0, x1, . . . , xn , it suffices to prove the identity for positive integers x0, x1, . . . , xn with x0 ≤ x1 ≤ · · · ≤ xn . We define a Vandermonde card to possess a suit and a value, where a card of Suit i has a value from the se...

The inversion of the Vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory. This work deals with the problem of getting an explicit formula for the generic element of the inverse. We derive two algorithms in O(n) and O(n) and compare them with the Parker–Traub and the Björck–Pereyra algorithms. 2005 Elsevier Inc. Al...

h our previous paper [l], we observed that generalized Vandermonde determinants of the form vn;~(ll,. , G) = \q” / I 1 5 i, k 5 n, where the 2, are distinct points belonging to an interval [a, b] of the real line, the index n stands for the order, the sequence p consists of ordered integers 0 5 pi < ~2 < ... < pLn, can be factored as a product of the classical Vandermonde determinant and a Schu...

In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems VΦ(x), i.e., polynomialVandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n2) computational procedure. We present numer...