Related papers: On discrete orthogonal polynomials of several vari…
We study the relation between certain non-degenerate lower Hessenberg infinite matrices $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known…
Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this…
We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…
In this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. In…
We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure…
In this survey article, we review some results and conjectures related to orthogonal polynomials on Cantor sets. The main purpose of this paper is to emphasize the role of equilibrium measures in order to have a general theory of…
Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in…
This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…
We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain…
We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the…
An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…
We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $\sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family $\{F_i \}$. The most important example is a polynomial with $c=1.$…
Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e…
Using methods developed in multivariate splines, we present an explicit formula for discrete truncated powers, which are defined as the number of non-negative integer solutions of linear Diophantine equations. We further use the formula to…
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.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…
Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…