q-正类型函数的性质

A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

We obtain an asymptotic expansion for the sum \[S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}\] as $a\rightarrow 0$ in $|\arg\,a|<π/2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with $m=0$. Numerical results are given to illustrate the accuracy of the expansion.

Uniqueness of positive solutions for boundary value problems associated with indefinite $ϕ$-Laplacian type equations

The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $ϕ$-Laplacian equation \begin{equation*} \bigl{(} ϕ(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} where $ϕ$ is a homeomorphism with $ϕ(0)=0$, $a(t)$ is a stepwise indefinite weight and $g(u)$ is a continuous function. When dealing with the $p$-Laplacian differential operator $ϕ(s)=|s|^{p-2}s$ with $p>1$, and the nonlinear term $g(u)=u^γ$ with $γ\in\mathbb{R}$, we prove the existence of a unique positive solution when $γ\in\mathopen{]}-\infty,(1-2p)/(p-1)\mathcl…

关于 LCA 群上正定函数的 Delsarte 型极值问题

On extremal problems of Delsarte type for positive definite functions on LCA groups

本文建立局部紧 Abel 群上 Delsarte 与 Turán 极值问题的统一框架，提出变体并分析表述等价性，证明一般拓扑条件下极值函数的存在性。

Korovkin-type Theorems and Approximation by Positive Linear Operators

This survey paper contains a detailed self-contained introduction to Korovkin-type theorems and to some of their applications concerning the approximation of continuous functions as well as of L^p-functions, by means of positive linear operators. The paper also contains several new results and applications. Moreover, the organization of the subject follows a simple and direct approach which quickly leads both to the main results of the theory and to some new ones.

Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type $BC$. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds $G/K$ over one of the (skew) fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H.$ We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hyper…

A Furstenberg-Katznelson-Weiss type theorem on (d + 1)-point configurations in sets of positive density in finite field geometries

We show that if $E \subset \mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, and $|E| \geq ρq^d$, where $ q^{-\frac{1}{2}}\ll ρ\leq 1$, then $E$ contains an isometric copy of at least $c ρ^{d-1} q^{d+1 \choose 2}$ distinct $(d+1)$-point configurations.

Existence and regularity of positive solutions to elliptic equations of Schrödinger type

We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schrödinger type, under only a form boundedness assumption on $σ\in D'(Ω)$ and ellipticity assumption on $\mathcal{A}\in L^\infty(Ω)^{n\times n}$, for an arbitrary open set $Ω\subseteq \mathbf{R}^n$. We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to certain elliptic equations with quadratic nonlinearity in the gradient. As a consequence, we obtain necessary and suff…

Weighted estimates for positive operators and Doob maximal operators on filtered measure spaces

We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero \cite{LaSaUr}. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen-Pérez type and Lerner-Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.

正测度集合的强型Furstenberg-Sárközy定理

A strong-type Furstenberg-Sárközy theorem for sets of positive measure

本文证明正测度平面集合中必存在一点，使得该集合与过此点的一整族幂型曲线非平凡相交，并证明参数区间在对数尺度上可任意大，推广了连续参数Sárközy型定理。

Positive intertwiners for Bessel functions of type B

Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\geq 0, k_2>0$. It was recently shown that the positivity of the operator $V_{k^\prime\!,k} =V_{k^\prime}\circ V_k^{-1}$ which intertwines the Dunkl operators associated with $k$ and $k^\prime=(k_1+h,k_2)$ implies that $h\in[k_2(n-1),\infty[\,\cup\,(\{0,k_2,\ldots,k_2(n-1)\}-\mathbb Z_+)$. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For $k_1 \…

正双对角分解与混合型切比雪夫多正交多项式

Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials

本文刻画了带状全正矩阵正双对角分解的存在性，通过Metelmann因子级数收敛性及Toeplitz符号负根给出全局分解，并推导混合型多正交多项式及其测度性质。