Related papers: Polynomial Valuations on Vector Lattices
We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are…
For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second…
Multivalued projections are applied to the study of weighted least squares solutions of linear relations equations (or inclusions) and some of its applications. To this end a matrix representation of multivalued projections with respect to…
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds…
We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlev\'e equations, in a Lax form,…
The motivation of this paper is to construct the theory of vector calculus of multivariate arithmetical functions. We prove analogues of integral theorems and Poincare's lemma.
We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of…
We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.
We present a complete classification of $\operatorname{SL}(n)$ contravariant, $C(\mathbb{R}^n\setminus\{o\})$-valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int.…
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
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…
We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are…
We prove martingale-ergodic and ergodic-martingale theorems with continuous parameter for vector valued Bochner integrable functions. We first prove almost everywhere convergence of vector valued martingales with continuous parameter. The…
We improve upon the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.
The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher…
Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer…
The theory of polynomials orthogonal with respect to one inner product is classical. We discuss the extension of this theory to multiple inner products. Examples include the Lam\'e and Heine-Stieltjes polynomials.
In this short lecture, we compute asymptotics of orthogonal polynomials, from a saddle point approximation. This is an example of a calculation which shows the link between integrability, algebraic geometry and random matrices.
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss…