Sharp van der Corput估计与最小均差

Relation between Turán extremum problem and van der Corput sets

Let $K\subset\mathbb N$ and $\mathbf T(K)$ is a set of trigonometric polynomials \[ T(x)=T_0+\sum_{k\in K, k\le H}T_k\cos(2πkx), \qquad H>1, \] $T(x)\ge0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le1/2$ and $K(h)$ is the class of functions \[ f(x)=\sum_{n=0}^{\infty}a_n\cos(2πnx) \] satisfying the following conditions: $a_n\ge0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le|x|\le1/2$. We consider an relation between extremum problem \[ δ(K)=\inf_{T\in\mathbf T(K)}T_0 \] and Turán extremum problem \[ A(h)=\sup_{f\in K(h)}a_0=\sup_{f\in K(h)}\int_{-h}^hf(x) dx \] for rational numbers $h=p/q$ a…

关于一致分布的一个准则

On a criterion of uniform distribution

本研究推广了van der Corput的序列一致分布准则，证明满足弱单调性及两个二阶差分条件的序列模1一致分布，采用有理数丢番图逼近的直接证法，区别于原准则的指数和估计方法。

差商

Divided Differences

本文通过定义差商的新定义，推导并讨论了单变量差商的基本性质和事实。

利用差商终止零平衡超几何级数

Terminating Zero-Balanced Hypergeometric Series Using Divided Differences

本文利用差商方法证明并推广零平衡终止超几何级数的求和公式，并推导出有限卷积变换。

指数函数差商的不等式、恒等式与界

Inequalities, identities, and bounds for divided differences of the exponential function

本文研究了指数函数差商的性质，证明其对数子模性，建立四点不等式，推导固定均值与方差下的精确双侧界及大输入渐近性，并给出卷积恒等式与重复参数求和公式等闭式恒等式。

Fourier transforms of irregular mixed homogeneous hypersurface measures

With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.

Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic

In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions.

Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices

In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients appearing in the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of the equation. In particular, we provide explicit expressions for the matrices in the three-term recurrence relations satisfied by the bivariate Racah polynomials introduced by Tratnik. Moreover, we present the family of monic bivariate Racah polynomials …

A new characterization of convexity with respect to Chebyshev systems

The notion of $n$th order convexity in the sense of Hopf and Popoviciu is defined via the nonnegativity of the $(n+1)$st order divided differences of a given real-valued function. In view of the well-known recursive formula for divided differences, the nonnegativity of $(n+1)$st order divided differences is equivalent to the $(n-k-1)$st order convexity of the $k$th order divided differences which provides a characterization of $n$th order convexity. The aim of this paper is to apply the notion of higher-order divided differences in the context of convexity with respect to Chebyshev systems int…

Characterizations of higher-order convexity properties with respect to Chebyshev systems

In this paper various notions of convexity of real functions with respect to Chebyshev systems defined over arbitrary subsets of the real line are introduced. As an auxiliary notion, a concept of a relevant divided difference and also a related lower Dinghas type derivative are also defined. The main results of the paper offer various characterizations of the convexity notions in terms of the nonnegativity of a generalized divided difference and the corresponding lower Dinghas type derivative.

A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the …

二维Hermite算子的尖锐端点本征函数估计

Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator

针对二维Hermite算子的谱投影，通过多种分析方法结合，证明了其从$L^2$到$L^{10/3}$的无对数尖锐端点估计，拓展了算子谱分析的相关结果。

Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidate…