Divided Differences
C. de Boor
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。
从差商的新定义出发，本文推导并讨论了（单变量）差商的基本性质和事实。
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.
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Divided Differences of Implicit Functions
研究隐函数y(x)由方程g(x,y)=0定义时，其分差用g的二元分差表示的问题，推广了逆函数分差的最新结果。
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。
Sharp van der Corput estimates and minimal divided differences
本文通过最小化均差节点和复均值定理，确定了子水平集估计及van der Corput引理中的尖锐常数，并改进了一般引理中的常数至渐近最优。
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。
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.
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 …
We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.
We study differentiability properties of functions defined in the euclidean space in terms of a conical square function which is analogue to the classical square function introduced by Stein and Zygmund in the sixties. Pointwise differentiability can be characterized, modulo sets of Lebesgue measure zero, in terms of the finiteness of the conical square function. At the complement of this set, a Law of the Iterated Logarithm describes the maximal growth of the divided differences in terms of a truncated version of the conical square function. Moreover a characterization of Sobolev spaces in te…
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…
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 …
We give an estimate of the general divided differences $[x_0,\dots,x_m;f]$, where some of the $x_i$'s are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen Whitney and Marchaud celebrated inequalities in relation to Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f\in C^{(r)}(I)$ and a set $Z=\{z_j\}_{j=0}^μ$ such that $z_{j+1}-z_j \geq λ|I|$, for all $0\le j \le μ-1$, where $I:=[z_0, z_μ]$, $|I|$ is the length of $I$ and $λ$ is some po…
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.
On the best constants of Schur multipliers of second order divided difference functions
本文基于双线性转移和Hörmander-Mikhlin-Schur乘子定理，给出了二阶差商函数双线性Schur乘子有界性的新证明，显著改进了其渐近界的上下界并证明了其在特定条件下的最优性。
On the best constants of Schur multipliers of higher order divided difference functions
本文研究了更高阶除差函数Schur乘子的最佳常数，通过分析多元Schur乘子的范数，得出了关于Hölder共轭和函数导数的估计，提供了Koplienko问题解决的关键定理的另一种证明。
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…
We consider the conjecture of Brutman and Pasow on a totality divided differences and prove the conjecture for continuous functions.
Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered.
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
We find the adjoint of the Askey-Wilson divided difference operator with respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy principle value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.
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