Discrete Lebedev-Skalskaya transforms

Discrete Kontorovich-Lebedev transforms

Discrete analogs of the classical Kontorovich-Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function $K_{in}(x), x >0, n \in \mathbb{N}, i $ is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application, a Dirichlet boundary value problem in the upper half-plane for inhomogeneous Helmholtz equation is solved.

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 Yor integral and a system of polynomials related to the Kontorovich-Lebedev transform

In this paper we establish different representations of the so-called Yor integral, which is one of the key ingredient in mathematical finance, in particular, to compute normalized prices of Asian options. We show, that the Yor integral is related with the Kontorovich-Lebedev transform. Also we discuss its relationship with a system of polynomials recently introduced by the author. We derive new important properties of these polynomials, including upper bounds, an exact asymptotic behavior for large values of their degree and explicit formula of coefficients.

The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_α$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $KL_α$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_α$-transform is a $d$-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular condition…

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…

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

Classical integral representation of the Mellin type kernel in terms of the Laplace integral gives an idea to construct a new class of non-convolution (index) transforms. Particular examples give the Kontorovich-Lebedev-like transformation and new transformations with hypergeometric functions as kernels. Mapping properties and inversion formulas are obtained. Finally we prove a new inversion theorem for the modified Kontorovich-Lebedev transform

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.

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。

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。

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.

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。

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。

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

We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion formulas are proved in a Banach space of functions whose Mellin's transforms are integrable over the vertical line Re s > 1. Examples of new transformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited. Particular cases include familiar Lambert and Mobius transformations. Finally a class of equivalences of the Salem type to the Riemann hypothes…

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

By using structural and asymptotic properties of the Kontorovich-Lebedev transform associated with Minkowski's question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc., 53 (3), (1943), p. 439) whether its Fourier-Stieltjes transform vanishes at infinity. This paper is substituted with a correct alternative and affirmative solution of Salem's problem and its generalization for derivatives of any order of the Fourier-Stieltjes transform of the Minkowski'i question mark function. The question is finally solved.