Discrete Kontorovich-Lebedev transforms

多项式Kontorovich-Lebedev变换下的中心因子

Central factorials under the Kontorovich-Lebedev transform of polynomials

我们表明，轻微的修改的Kontorovich-Lebedev变换导致多项式的向量空间的自同构。这种情况以及修改的贝塞尔函数的梅林变换性质执行单项式到中心阶乘多项式的通道。一个特别的注意力被驱动到多项式序列的KL变换是典型的序列，这将是充分的特点。最后，得到了中心因子与欧拉多项式之间的新的恒等式.

与傅里叶余弦-正弦变换和Kontorovich-Lebedev变换相关的三角加权广义卷积算子

Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations

本研究引入带三角权重γ=sin y、关联傅里叶余弦-正弦与Kontorovich-Lebedev变换的广义卷积算子，证明其在双参数Lebesgue空间的有界性，给出加权L_p范数估计并得到相关卷积积分微分方程的L_1可解条件。

由Mellin和Laplace算子生成的一类指数变换

A class of index transforms generated by the Mellin and Laplace operators

经典的整表示的梅林型核的Laplace积分给出了一个想法，构造一类新的非卷积（指数）变换。特别的例子给出了Kontorovich-Lebedev变换和新的变换与超几何函数作为内核。得到了映射性质和反演公式。最后证明了修正Kontorovich-Lebedev变换的一个新的反演定理

Discrete Lebedev-Skalskaya transforms

Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{α+in}(x), {\rm Im} K_{α+in}(x), x >0, n \in \mathbb{N}, |α| < 1,\ i $ is the imaginary unit and $K_ν(z)$ is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when $α= \pm 1/2$. The case $α=0$ reduces to the Kontorovich-Lebedev transform.

New index transforms of the Lebedev- Skalskaya type

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second order partial differential equation, involving the Laplacian, is obtained. It is noted, that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich- Lebede…

与指标Kontorovich-Lebedev变换和Fourier积分相关的多重卷积的复合结构

Composition structure of polyconvolution associated with index Kontorovich-Lebedev transform and Fourier integrals

本文基于Kakichev概念和Yakubovich-Britvina方法，提出与Kontorovich-Lebedev变换及Fourier积分相关的新型多重卷积结构，证明酉性条件、Plancherel型定理及加权$L_p$不等式，并应用于Toeplitz-Hankel方程求解。

某些指标核的一致上界与渐近展开

Uniform upper bounds and asymptotic expansions for certain index kernels

本文为 Kontorovich-Lebedev 算子相关的指标变换核建立一致上界和渐近展开，涵盖 Mehler-Fock、Lebedev、指标 Whittaker 及 Olevskii 变换。

New index transforms with the product of Bessel functions

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.

Whittaker函数的一般化乘积公式及一类新型指标变换

Generalized product formulas for Whittaker's functions and a novel class of index transforms

本文建立了涉及不同指标Whittaker函数乘积的一般化乘积公式和指标变换，并给出反演公式，涵盖修正Bessel函数等特殊情形，利用复指标Kontorovich-Lebedev和Olevskii变换。

Discrete Lebedev's index transforms

Discrete analogs of the Lebedev transforms with the product of the modified Bessel functions are introduced and investigated. Several expansions of suitable functions and sequences in terms of the series and integrals, involving the modified and incomplete Bessel functions are established.

基于Ramanujan恒等式和Salem型等价的积分变换和级数变换到黎曼假设

Integral and series transformations via Ramanujan's identities and Salem's type equivalences to the Riemann hypothesis

我们考虑积分和系列变换，这是与拉马努金的身份，涉及各种算术函数和黎曼的zeta函数的不同参数的产品的比率。证明了Mellin变换在垂直线Re s > 1上可积的函数的巴拿赫空间中的互逆反演公式.新的转换，如Widder-Lambert和Kontorovich-Lebedev型的例子展出。特别的例子包括大家熟悉的兰伯特和Mobius变换最后建立了Salem型与黎曼假设的一类等价。

Lebedev's type index transforms with the squares of the associated Legendre functions

The classical Lebedev index transform (1967), involving squares and products of the Legendre functions is generalized on the associated Legendre functions of an arbitrary order. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. As an interesting application, a solution to the boundary value problem for a third order partial differential equation is obtained.

Lebedev's type index transforms with the modified Bessel functions

New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the mod…

On the generalized Lebedev index transform

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel $|K_{(iτ+α)/2}(x)|^2, α\ge 0,\ x >0, \ τ\in \mathbb{R},$ where $K_μ(z)$ is the Macdonald function and $ i$ is the imaginary unit. Mapping properties such as the boundedness, compactness, invertibility are investigated for these operators and their adjoints in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. As an interesting application, a solution o…

Whittaker函数平方的新指标变换

A new index transform with the square of Whittaker's function

本文提出并研究了一种基于Whittaker函数平方的指标变换，建立了反演公式，其特例涵盖Lebedev型变换。

闵可夫斯基的傅里叶-斯蒂尔切斯变换？(x)函数和对Salem问题的肯定回答

The Fourier-Stieltjes transform of Minkowski's ?(x) function and an affirmative answer to Salem's problem

利用与Minkowski问号函数相联系的Kontorovich-Lebedev变换的结构和渐近性质，我们对R. Salem（美国数学学会，53（3），（1943），p.439）是否其傅里叶-斯蒂尔切斯变换在无穷远处消失。本文给出了Salem问题的一个正确的选择和肯定的解，并将其推广到Minkowski问号函数的傅里叶-斯蒂尔切斯变换的任意阶导数上。问题终于解决了。