Julius Borcea, Alexander Guterman, Boris Shapiro
Following the classical approach of Pólya-Schur theory we initiate in this paper the study of linear operators acting on $\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.
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We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.
The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a finite number of simpler bifurcations, and these bifurcations are characterized.
Shape preserving approximation of periodic functions -- Conclusion
本文给出周期函数三角多项式共单调逼近的Jackson型估计最终结果，总结保形逼近研究，并指出其与代数多项式情形有本质差异。
We introduce the -1 dual Hahn polynomials through an appropriate $q \to -1$ limit of the dual q-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical orthogonal polynomials of the Askey scheme, the -1 dual Hahn polynomials do not exhibit the Leonard duality property. Instead, these polynomials satisfy a 4-th order difference eigenvalue equation and thus possess a bispectrality property. The corresponding generalized Leonard pair consists of two matrices $A,B$ each of size $N+1 \times N+1$. In the eigenbasis wher…
It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $q\ge 3$ is completely different from comonotone and coconvex …
Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials
本文综述了近三十年来有限区间上代数多项式保形逼近理论的发展，讨论了均匀和逐点估计的有效性与无效性，并比较了不同约束和无约束逼近空间及特定函数的逼近阶。
It is well known that the family of Hahn polynomials $\{h_n^{α,β}(x;N)\}_{n\ge 0}$ is orthogonal with respect to a certain weight function up to $N$. In this paper we present a factorization for Hahn polynomials for a degree higher than $N$ and we prove that these polynomials can be characterized by a $Δ$-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials and give the limit relations between them for all $n\in \XX N_0$. Furthermore, in order to get this results for the Krawtchouk polynomials we will get a more general property of orthogo…
A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed d…
The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilized to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g., the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular different…
Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels
该研究建立实直线紧子集容许最优多项式网格的位势论充分条件，结合Andrievskii估计证明一致完美紧子集存在最优多项式网格，经乘积论证得平面康托尔尘集上的最优网格。
We apply Rossi's half-plane version of Borel's Theorem to study the zero distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and differential operators (with entire coefficients) preserving subsets of $\mathcal{A}$-entire functions, and hence obtain several analogous results for the Hermite-Poulain Theorem to linear finite ($q$-)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomi…
This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC_2 a new pair of lowering and raising operators is obtained.
On sequences preserving summability
该研究聚焦保持可和性的正实数序列，证明其序列族的开集性质，并介绍了多项式、幂和、部分差分方程解等多类保持可和性的序列。
Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
本文研究由实正交多项式系定义的线性变换，通过其微分算子表示中的系数多项式根的交错性质，刻画了保持实根性的条件，并证明相关猜想。
We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.
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