Related papers: Biorthogonal polynomials and zero-mapping transfor…
We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of…
We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings…
We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $\sigma (P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$.…
Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…
Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the…
We consider polynomials orthogonal on the unit circle with respect to the complex-valued measure $z^{\omega-1}\mathrm{d} z$, where $\omega\in\mathbb{R}\setminus\{0\}$. We derive their explicit form, a generating function and several…
This note is about the observation that the various transition formulas between bases of trigonometric polynomials can be expressed in terms binomial coefficients. More specifically, we write the entries of the Chebyshev matrices $ T$ and $…
We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant…
It turned out that the partial sums $g_n(z) = \sum_{k=0}^n \frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k} \frac{z^k}{k!}$, of the generalized hypergeometric series ${}_p F_q(a_1,...,a_p; b_1,...,b_q;z)$, with parameters…
We propose a new algorithm for generating pseudorandom (pseudo-generic) numbers of conformal measures of a continuous map T acting on a compact space X and for a Holder continuous potential F. In particular, we show that this algorithm…
We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…
We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…
Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained…
We prove that for any Borel probability measure $\mu$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x…
We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the…
For a system of two measures supported on a starlike set in the complex plane, we study asymptotic properties of associated multiple orthogonal polynomials $Q_{n}$ and their recurrence coefficients. These measures are assumed to form a…
The main aim of this work is to apply the study of the asymptotic behaviour of generalized eigenvalues between infinite Hermitian definite positive matrices in an important question regarding the location of zeros of Sobolev orthogonal…
We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator modulo the sigma-ideal generated by wandering sets. This implies that if the entropies of invariant…
Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and…
We show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for…