A direct computation of a certain family of integrals

多参数傅里叶积分算子的一族

A multi-parameter family of Fourier integral operators

本文研究了一类多参数傅里叶积分算子，证明其将H^1-霍尔德空间映射到L^1空间，并获得精确的L^p估计，同时引入了广义索博列夫空间并应用到波动方程的先验估计中。

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which m…

A new family of sharp conformally invariant integral inequalities

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.

一类多对数积分

A family of polylogarithmic integrals

针对统计等离子体物理中电子熵 Sommerfeld 温度展开出现的含参多对数积分，本文给出了以特殊函数表示的完整闭式描述。

On a Family of Integrals that extend the Askey-Wilson Integral

We study a family of integrals parameterised by $ N = 2,3,\dots $ generalising the Askey-Wilson integral $ N=2 $ which has arisen in the theory of $q$-analogs of monodromy preserving deformations of linear differential systems and in theory of the Baxter $Q$ operator for the $ XXZ $ open quantum spin chain. These integrals are particular examples of moments defined by weights generalising the Askey-Wilson weight and we show the integrals are characterised by various $ (N-1) $-th order linear $q$-difference equations which we construct. In addition we demonstrate that these integrals can be eva…

A new family of singular integral operators whose $L^2$-boundedness implies rectifiability

Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}^1(E)<\infty$. David and Léger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\, z/|z|^2$, $z\in \mathbb{C}\setminus\{0\}$) has the following property $(*)$: the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property…

On the $\ell^s$-boundedness of a family of integral operators

In this paper we prove an $\ell^s$-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal $L^p$-regularity for parabolic problems with time-dependent generator is developed.

Master Theorems for a Family of Integrals

A family of general Master theorems for analytic integration over the real (or imaginary) axis with various reciprocal hyperbolic (trig) kernels ($\sinh and/or \cosh$) with varying arguments is developed. Several examples involving closed-form integrations which do not appear to exist in the standard tables are given in detail and general results are listed in an Appendix. As well, it is shown how to convert the special case of an infinite series involving ratios of Gamma functions into a finite series.

Weighted variation inequalities for differential operators and singular integrals

We prove weighted strong $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators. For the first family of operators the weights used can be either Sawyer's one-sided $A^+_p$ weights or Muckenhoupt's $A_p$ weights according to that the differential operators in consideration are one-sided or symmetric. We use only Muckenhoupt's $A_p$ weights for the second family. All these inequalities hold equally in the vector-valued case, that is, for functions with values in $\el^ρ$ for $1<ρ<\infty$. As application, we show variation inequalities for mean bounded posit…

Regularity of multi-parameter Fourier integral operator

We study a family of Fourier integral operators, by allowing their symbols to satisfy a multi-parameter differential inequality. We extend the sharp L^p-result obtained by Seeger, Sogge and Stein to product spaces.

A family of singular oscillatory integral operators and failure of weak amenability

A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$ dimensional Heisenberg group and $SL(2,\Bbb R)$ acts via the irreducible representation of dimension $2n$ fixing the center of $H_n$. We show that these groups fail to be weakly amenable. Following an idea of Haagerup for the case $n=1$ one can reduce matters to the problem of obtaining nontrivial uniform bounds for a family of singular oscillatory integral operators…

Self-reciprocal functions and double Mordell integrals

The theory of self-reciprocal functions is applied to the study Mordell type integrals. We find two particular eigenfunctions of the double cosine Fourier transform and then use them to evaluate certain one- and two-dimensional Mordell type integrals in closed form. A reduction formula is given for a certain family of double Mordell integrals in terms of one-dimensional integrals.

A family of singular integral operators which control the Cauchy transform

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-bo…

Hua type beta-integrals and projective systems of measures on flag spaces

We construct a family of measures on flag spaces (or, equivalently, on the spaces of upper-triangular matrices) compatible with respect to natural projections. We obtain an $n(n-1)/2$-parametric family of beta-integrals over space of upper-triangular matrices of size $n$.

Existence and computation of Riemann-Stieltjes integrals through Riemann integrals

We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we present a new generalization to the well-known formula for continuous integrands and continuously differentiable integrators.

The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of Logarithms and Exponentials

We present the evaluation of a family of exponential-logarithmic integrals. These have integrands of the form P(exp(x),ln(x)) where P is a polynomial. The examples presented here appear in sections 4.33, 4.34 and 4.35 in the classical table of integrals by I. Gradshteyn and I. Ryzhik.

Numerical computation of real or complex elliptic integrals

Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.

Degree of the First Integral of a Foliation in the Pencil $\mathcal{P}_4$

Let $\mathcal{P}_4$ be the linear family of foliations of degree 4 in $\mathbb{P}^2$ given by A. Lins Neto, whose set of parameters with first integral $I_p(\mathcal{P}_4)$ is dense and countable. In this work, we will calculate explicitly the degree of the rational first integral of the foliations in this linear family, as a function of the parameter.

A family of Exponential Integrals suggested by Stellar Dynamics

While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions $$ I(λ,μ,ν; z) :=\int_0^zx^λ\,\Enu(x^μ)\,dx= {γ\left({1+λ\overμ},z^μ\right) + z^{1+λ}\Enu(z^μ)\over 1+λ+ μ(ν-1)}, \quad μ>0,\quad z\geq 0, \eqno (1) $$ whe…

傅里叶及其超越：一族积分变换的不变性性质

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

本文提出傅里叶变换可由伸缩交织关系和高斯特征函数唯一刻画，并由此构造一族类傅里叶积分变换，证明其酉性、周期性、特征值以及短时版本的重构与正交性质。

参数双线性分数阶积分的一致有界性

Uniform boundedness of parametric bilinear fractional integrals

本文研究欧拉-里斯系统中出现的双线性分数阶积分族，给出了关于自然参数的一致弱型有界性，并证明该界是尖锐的。

Singular integral equations with applications to travelling waves for doubly nonlinear diffusion

We consider a family of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for a family of diffusion-convection-reaction equations involving the $p$-Laplacian operator. Our results extend the ones due to B.\,Gilding for the case $p=2$. The fact that $p\neq2$ modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools for the analysis. The results for the integral equation are then used to study the existence and properties of travelling-wave solutions for doubly nonlinear diffusion-reaction …

A General Integral Identity

A family of general integral identities is derived and several applications of physical interest are presented

Fourier Transform Representation of the Extended Fermi-Dirac and Bose-Einstein Functions with Applications to the Family of the Zeta and Related Functions

On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated befo…

On some integrals involving the Hurwitz zeta function: part 2

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin (πq), ln Gamma(q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A_k(q):=k zeta'(1-k,q), where k is a natural number, and a family of polygamma functions of negative order, whose properties we study in some detail.

Computation of a Feynman integral

A Feymnan integral is computed exactly using LLL

On a Family of Hypergeometric Polynomials

We work on the SCE problems. We establish the expressions of three integrals' sequences, related to it, in terms of five families of polynomials. Relations between these integrals are demonstrated and we focus on one of the three problems : the determination of the family of polynomials noted $e_n (n \in \mathbb{N})$. We show taht these polynomials are hypergeometric. From this property, the NU method can be applied to this family. We have been able to determine the Rodrigues formula. These polynomials have properties that distinguish them from classical hypergeometric polynomials. We state an…