相关论文: Incomplete Symmetric Orthogonal Polynomials of Fin…

We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the…

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of…

In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It is shown that such a polynomial…

In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…

The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the…

We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this classification is a list of equations containing no…

The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear…

The rules to write out any one of the linearly independent functions belonging to the infinite set of those in polynomial form that satisfy the sourceless Grad-Shafranov equation as stated in the toroidal-polar coordinate system are…

We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

In this paper, we will discuss the notion of almost orthogonality in a functional sequence.Especially, we will define a few sequences of almost orthogonal polynomials which can be used successfully for modeling of electronic systems which…

Let $K$ be a field and let $\mathbb N = \{1,2, \dots \}$. Let $R_n=K[x_{ij} \mid 1\le i\le n, j\in \mathbb N]$ be the ring of polynomials in $x_{ij}$ $(1 \le i \le n, j \in \mathbb N)$ over $K$. Let $S_n = Sym (\{1,2, \ldots, n \})$ and…

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality…

We introduce and study symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $\Delta=\pm 1/2$. There is also a close relation to…

We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…

We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact"…

In this paper, we will establish a general method of studying finite-dimensional normed spaces, and apply this method to classifying $3$-dimensional and $4$-dimensional normed spaces over a non-spherically complete field. For this purpose,…

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular,…

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation.…