相关论文: Solution of the polynomial moment problem
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\phi$ on…
Given a parametrised weight function $\omega(x,\mu)$ such that the quotients of its consecutive moments are M\"obius maps, it is possible to express the underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present paper…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The classical Center-Focus Problem posed by H. Poincar\'e in 1880's is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\dfrac{\partial}{\partial x}+(x+Q(x,y))\dfrac{\partial}{\partial y},$ with…
The paper is devoted to quantization of polynomial momentum observables in the cotangent bundle of a smooth manifold. A quantization procedure is proposed allowing to quantize a wide class of functions which are polynomials of any order in…
We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the…
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
We propose a moment relaxation for two problems, the separation and covering problem with semi-algebraic sets generated by a polynomial of degree d. We show that (a) the optimal value of the relaxation finitely converges to the optimal…
Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials…
We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization…
In this note I collect some typical examples of orthogonal polynomials with simple moments where both moments and orthogonal polynomials have nice q-analogs.
We deal with and investigate sparse univariate Positivstellens\"atze, Nichtnegativstellens\"atze, and solutions to sparse moment problems. The paper relies heavily on results on T-system by Karlin in 1963 and by Karlin and Studden in 1966.…
We attempt to get a polynomial solution to the inverse problem, that is, to determine the form of the mechanical Hamiltonian when given the energy spectrum and transition dipole moment matrix. Our approach is to determine the potential in…
In this paper, we devote our interest to solving the real cubic truncated moment problem. We provide some results that allow to get a complete solution via a minimal representing measure. Some numerical examples are also presented to…
Orthogonal q-polynomials associated with q-Laguerre-Hahn form will be studied as a generalization of the q-semiclassical forms via a suitable q-difference equation. The concept of class and a criterion to determinate it will be given. The…
Given $n$ point masses turning in a plane at a constant speed, this paper deals with the global bifurcation of periodic solutions for the masses, in that plane and in space. As a special case, one has a complete study of n identical masses…