On an integral as an interval function

Some New Generalized Results on Ostrowski Type Integral Inequalities With Application

The aim of this paper is to establish some new inequalities similar to the Ostrowski's inequalities which are more generalized than the inequalities of Dragomir and Cerone. The current article obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval. Some new purterbed results are obtained. Application for cumulative distribution function is also discussed.

Interval Functions and their Integrals, by Ralph Henstock, Ph.D. thesis, 1948

Ralph Henstock (1923 - 2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950's. As a Cambridge undergraduate Henstock took a course of lectures, by J.C. Burkill, on the integration of interval functions. Later, under the supervision of Paul Dienes in Birkbeck College, London, he undertook research into the ideas of Burkill (interval function integrands) and Dienes (Stieltjes integrands); and he presented this thesis in December 1948. The thesis contains the germ of Henstock's later work, in…

On The Generalisation of Henstock-Kurzweil Fourier Transform

In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is also shown that this integral is more general than the Henstock-Kurzweil integral. Finally, the Fourier transform is defined using the Laplace integral, and its well-known properties are established.

$x^m (-\log x)^l$ 在区间 $0\le x \le 1$ 内的 Chebyshev 逼近

Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$

本文研究了 $x^m (-\log x)^l$ 关于平移 Chebyshev 多项式的级数展开，通过分部积分将所需积分族化简为已知可由 polygamma 函数有限和表示的 $m=0$ 积分，解决了相关计算问题。

A Simple Multiple Integral Solution to the Broken Stick Problem

Regard the closed interval $[0,1]$ as a stick. Partition $[0,1]$ into $n+1$ different intervals $I_1, \ \dots \ , I_{n+1},$ where $n \geq 2,$ which represent smaller sticks. The classical Broken Stick problem asks to find the probability that the lengths of these smaller sticks can be the side lengths of a polygon with $n+1$ sides. We will show that this probability is $1-\frac{n+1}{2^{n}}$ by using multiple integration.

Generalized Stieltjes constants and integrals involving the log-log function: Kummer's Theorem in action

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function $x\mapsto \ln \ln(1/x)$.

Integral averages and maximization of functions

Some properties of integral averages of functions on intervals and their asymptotic behavior are investigated. The results are aimed at applications to entire and subharmonic functions.

On Approximation Properties for Non-linear Integral Operators

We investigate the problem of pointwise convergence of the family of non-linear integral operators: \begin{equation} L_λ(f,x) = \int_a^b \sum_{m=1}^N f^m(t) K_{λ,m}(x,t) dt, \end{equation} where $λ$ is a real parameters, $K_{λ,m}(x,t)$ is non-negative kernel and $f$ is the function in $L_{1}(a,b)$. We consider two cases where $% (a,b)$ is a finite interval and when is the whole real axis.

An "algebraic" reconstruction of piecewise-smooth functions from integral measurements

This paper presents some results on a well-known problem in Algebraic Signal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier coefficients, Radon transform, etc.). Our results concern reconstruction (from the moments) of signals in two specific classes: linear combinations of shifts of a given function, and "piecewise $D$-finite functions" which satisfy on each continuity interval a linear differential equation with polynomial coefficients.

An integral that counts the zeros of a function

Given a real function $f$ on an interval $[a,b]$ satisfying mild regularity conditions, we determine the number of zeros of $f$ by evaluating a certain integral. The integrand depends on $f, f'$ and $f''$. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of $f$ by evaluating finitely many values of $f,f'$ and $f''$. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.

有界区间上振荡积分的高斯求积规则

A Gaussian quadrature rule for oscillatory integrals on a bounded interval

本文研究区间[-1,1]上带振荡权函数e^{iωx}的高斯求积规则及其正交多项式，证明该规则具有高渐近阶，且对所有ω保持精度，当ω→0时退化为经典Gauss-Legendre规则。

Inverses of gamma functions

Euler's Gamma function $Γ$ either increases or decreases on intervals between two consequtive critical points. The inverse of $Γ$ on intervals of increase is shown to have an extension to a Pick-function and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and in particular integral representations for the double gamma function and the $G$-function of Barnes are found.

On the Skewed Fractional Diffusion Advection Reaction Equation on the Interval

This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection reaction equation (FDARE) with variable coefficients on the bounded interval. The existence of the true (classical) solution together with norm estimation is established and the precise regularity bound is found; also, the structure of the solution is unraveled, capturing the essence of regularity, singularity, and other features of the solution. The key analy…

带Robin边界条件的一维Poisson方程解的积分均值

Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions

本文研究带Robin边界条件的一维Poisson方程，利用函数极化方法证明解的凸积分均值不等式，找到最大凸积分均值的解并证明其唯一性。

On Montgomery's pair correlation conjecture: a tale of three integrals

We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function $F(α, T)$ in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery's pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known …

Oscillation of Fourier Integrals with a spectral gap

Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.

临界区间内的 $L_p$ 弦 Minkowski 问题

The $L_p$ Chord Minkowski problem in a critical interval

本文在无对称性假设下，解决了 $0\leq p<1$ 区间内的 $L_p$ 弦 Minkowski 问题，该问题推广了过去 20 年广泛研究的 $L_p$ Minkowski 问题。

Bochner积分与函数值凸组合之差的范数估计

Norm Estimates for the Difference Between Bochner's Integral and the Convex Combination of Function's Values

本文针对具有Radon-Nikodym性质的Banach空间中的向量值函数，给出了Bochner积分与函数值凸组合之差的范数估计。

Macdonald函数与Gegenbauer函数的广义积分

Generalized integrals of Macdonald and Gegenbauer functions

本研究针对Macdonald与Gegenbauer函数双线性积分仅有限参数范围收敛的问题，采用可拓展标准积分的广义积分方法，通过解析延拓得到一般参数下的积分结果，还得到整数参数下与标度不变性反常相关的显式附加项。