Related papers: Approximation and orthogonality on fully symmetric…
We study the following problem: describe the triplets $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}{\rho}\sum_{ij} \partial_i…
Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system…
A contragenic function in a domain $\Omega\subseteq\mathbf{R}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic function, which is orthogonal in $L^2(\Omega)$ to all monogenic functions and their…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$…
In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : \theta \mapsto \prod_{1\le j \le M} \vert 1 - e^{i(\theta_{j}-\theta)}\vert…
For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a…
We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…
A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the…
We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline…
A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of…
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex…
We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally…
Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
The symmetric group on 4 letters has the reflection group $D_{3}$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_{3}$ and $D_{3}$. The isomorphism is used to construct an orthogonal basis of…
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a…