On the remainder in the Taylor theorem
Lior Bary-Soroker, Eli Leher
本文给出了泰勒定理中余项的界的一个简单明了的证明。证明只使用归纳法和事实，$f'\geq 0$意味着$f$的单调性，所以它可能是一个有吸引力的证明给本科生。
We give a short straightforward proof for the bound of the reminder term in the Taylor theorem. The proof uses only induction and the fact that $f'\geq 0$ implies the monotonicity of $f$, so it might be an attractive proof to give to undergraduate students.
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Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral
当一个实值函数的一个变量是近似的$n^{th}$度泰勒多项式，其余的估计使用Alexiewicz和Lebesgue $p$ -范数的情况下，$f^{(n)}$或$f^{(n+1)}$是Henstock-Kurzweil可积。当唯一的假设是$f^{(n)}$是Henstock-Kurzweil可积的，然后使用修改形式的$n^{th}$度泰勒多项式。当唯一的假设是$f^{(n)}\in C^0$，则通过将Alexiewicz范数应用于1阶Schwartz分布来估计余数。
Inequalities for the exponential remainder of the Taylor series
这是1992年的预印本，有一些更新。研究了指数函数泰勒级数的截面。G.Hardy，Kesava Menon，W. Gautschi，H.Alzer and others.本预印本的主要目的是研究具有最佳常数的主要不等式及其多重推广的新证明。一些公式化（其中一些最近被证明，见2016年的评论）。
We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $Π$-rough paths for $Π=\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that $p_{i}=pk_{i}^{-1}\ $for some $k_{i}\in \left\{ 1,\dots , \left[ p\right] \right\} $, we obtain a refined Taylor remainder estimate that contains a factorial decay component. The remainder estimates are in the right order as they are comparable to the next term in the Taylor expansion.
Uniform Factorial Decay Estimate for the Remainder of Rough Taylor Expansion
建立了受控微分方程解的Taylor逼近的一致阶乘衰减估计。它的证明需要一个阶乘衰减估计的控制路径，这是有趣的本身。
Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel
本文证明了一个非终止的适定基本 q-Taylor 展开，该展开具有互补余项，适用于第二作者在文献中隐含提出的双基无穷乘积核，并利用适定 Cooper 公式、Jackson 终止型求和、Rogers 求和以及 theta 插值完成证明，同时恢复了 Bailey 非终止求和作为推论。
This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error est…
Exponential Taylor Series
本文提出将可微复值函数表示为 $(1-e^{λx})$ 幂次之和的方法，给出余项显式公式，证明在适当条件下无穷级数可恢复 $C^\infty$ 原函数，并应用于计算涉及 Stirling Numbers 的无穷级数。
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.
On Ikehara type Tauberian theorems with $O(x^γ)$ remainders
出于解析数论，我们探讨剩余版本的池原的Tauberian定理产生幂律剩余条款。更准确地说，对于$f:[1,\infty)\rightarrow{\mathbb R}$非负和非减，我们证明了$f(x)-x=O(x^γ)$与$γ<1$在一定的假设下对$f$。我们提出了一个关于最弱的自然假设的猜想，并表明我们不能期望更多。
Direct and inverse approximation theorems of $2π$-periodic functions by Taylor--Abel--Poisson means
在积分度量下，得到了Taylor Abel Poisson算子对$2π$ -周期函数的正、逆逼近定理.
Approximation theorems for multivariate Taylor-Abel-Poisson means
在积分度量中，利用Taylor-Abel-Poisson平均得到了多元函数的正逼近和逆逼近定理。证明了空间$L_{p,Y}(\mathbb T^d)$中乘子的范数对所有正整数$d.$都是等价的
Taylor's formula and related inequalities for a derivative with a new parameter
本文给出了β-分数阶导数的泰勒定理。我们还研究了泰勒定理的一些新性质以及与此导数有关的一些有用的定理。我们将一些最近的和经典的积分不等式推广到这个新的简单有趣的分数微积分，包括Steffensen和Hermit-Hadamard不等式。
We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).
Double-sided Taylor's approximations and their applications in Theory of analytic inequalities
本文研究了双边泰勒逼近。本文给出了关于双边泰勒逼近的一个著名定理的简短证明。此外，证明了两个新的定理关于这种逼近的单调性。然后给出了双边泰勒逼近在解析不等式理论中的一些新应用。
Note on the absence of remainders in the Wiener-Ikehara theorem
证明了即使假定Mellin变换在减去半平面极点后的解析延拓，也不可能得到比Wiener-Ikehara定理中的经典余项更好的余项.我们还证明了一个类似的结果的Ingham-Karamata定理。
On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications
本文证明了复指数$x\in\R$的泰勒展开式中余项的一个新的界，给出了概率分布的特征函数（傅里叶-斯蒂尔切斯变换）的泰勒展开式的首项的近似精度的矩型估计。此外，还利用绝对三阶矩建立了概率分布的三阶矩的精确上界，使Jensen不等式更加尖锐.在此基础上，得到了特征函数及其导数在前三阶矩固定的分布类中{ \it uniform}的新的改进界.
We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set $C_1 \times C_2$ utilizing the notion of thickness. Specifically, we show that $C_1, C_2 \subset \mathbb{R}^d$ compact sets with thickness satisfying $τ(C_1) τ(C_2) >1$, then the edge lengths in $C_1 \times C_2$ corresponding to any pinned finite tree configuration has non-empty interior. Originally proven for Cantor sets on the real line by McDonald and Taylor, we use the notion of thickness introduced by Falconer and Yavicoli which allows us to generalize the result of McDonald and …
A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator
本文利用E. Askey-Wilson均差算子的Rains椭圆延拓我们考虑的扩展定理扩展了M.E.H. Ismail的Askey-Wilson单项基展开式三个直接的应用（基本上已经由于雨）包括简单的证明弗伦克尔和Turaev的椭圆扩展杰克逊的8-phi-7求和和贝利的10-phi-9变换，并计算连接系数Spiridonov的椭圆扩展拉赫曼的双正交有理函数。我们预示其他的例子，包括杰克逊的8-phi-7求和和二次扩展的无限扩展。
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space…
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