Kontorovich Lebedev Transform Research Articles

The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this ‐transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d‐orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose ‐transform is a d‐orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.

Multidimensional Kontorovich–Lebedev transforms

Different approaches are used to introduce a direct product Kontorovich–Lebedev transformation, its essentially multidimensional analogue and certain modifications in terms of multiple integrals over ℝ n +. Mapping properties are investigated. Inversion formulas are proved.

A class of polynomials and discrete transformations associated with the Kontorovich–Lebedev operators

We consider a class of polynomials related to the kernel K iτ(x) of the Kontorovich–Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoulli's and Euler's numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylor's series in terms of the Kontorovich–Lebedev operator.

HARMONIC ANALYSIS OF THE LEBEDEV-STIELTJES INTEGRALS

We expand the Bochner technique on the following Lebedev-Stieltjes integrals [Formula: see text] which are related to the Kontorovich-Lebedev transformation. Mapping and inversion properties are investigated. The Fourier type series with respect to an uncountable orthonormal system of the modified Bessel functions are considered in the Bohr type pre-Hilbert space. The Bessel inequality and Parseval equality are proved.

On a progress in the Kontorovich–Lebedev transform theory and related integral operators

This survey represents mainly recent results of the author, which were obtained during the last decade and concern the theory of the Kontorovich–Lebedev transformation and some related integral operators of the convolution and non-convolution type. In particular, we discuss their boundedness in the Lebesgue, Hardy and Sobolev type spaces and in some spaces of distributions. Special attention is given to the L 2-case and Plancherel's theory for these operators and to the harmonic analysis of the so-called Lebedev–Stieltjes integrals. Moreover, we concentrate on asymptotic, summation formulas and uncertainty principles for the Kontorovich–Lebedev transform.

Asymptotic and summation formulas related to the Lebedev integrals

We establish asymptotic and summation properties of the Lebedev integrals with respect to an index of the modified Bessel function, which are related to the known Kontorovich–Lebedev transformation. Analogues of the Watson lemma and Poisson summation formulas are proved. As applications certain type series involving Euler's gamma-functions and hyperbolic functions are evaluated.

The Kontorovich–Lebedev integral transformation with a Hankel function kernel in a space of generalized functions of doubly exponential descent

A version of the Kontorovich–Lebedev transformation with the Hankel function of second kind in the kernel is investigated in a space of distributions of doubly exponential descent. The inversion theorem is rigorously established making use in some steps of the proof of a relation of this transform with the Laplace one. Finally, the theory developed is illustrated in solving certain type of partial differential equations.

On the least values of [formula omitted]-norms for the Kontorovich–Lebedev transform and its convolution

We establish analogs of the Hausdorff–Young and Riesz–Kolmogorov inequalities and the norm estimates for the Kontorovich–Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in L p spaces, 1 ⩽ p ⩽ ∞ . Boundedness properties of the Kontorovich–Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich–Lebedev operator K i τ : L p ( R + ; x dx ) → L p ( R + ; x sinh π x dx ) , 2 ⩽ p ⩽ ∞ K i τ [ f ] = ∫ 0 ∞ K i τ ( x ) f ( x ) dx , τ ∈ R + is equal to π 2 1 - 1 p . It confirms, for instance, by the known Plancherel-type theorem for this transform when p = 2 .

Integral representation of one-dimensional three particle scattering for δ function interactions

The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.

On an Asymptotic Expansion of the Kontorovich–Lebedev Transform

An asymptotic expansion valid for large positive values of s is constructed for the integral transform where denotes the modified Bessel function of purely imaginary order s and the function f(x) is analytic in a suitable sector containing the positive real axis and possesses the asymptotic expansion as x→ ∞ in this sector, where γ is complex; and .

Relation Between Relativistic and Non-Relativistic Quantum Mechanics as Integral Transformation

A formulation of quantum mechanics (QM) in the relativistic configurational space (RCS) is considered. A transformation connecting the non-relativistic QM and relativistic QM (RQM) has been found in an explicit form. This transformation is a direct generalization of the Kontorovich–Lebedev transformation. It is shown also that RCS gives an example of non-commutative geometry over the commutative algebra of functions.

A Simplified Asymptotic Expansion of the Kontorovich-Lebedev Transform

A Simplified Asymptotic Expansion of the Kontorovich-Lebedev Transform

On the Theory of the Kontorovich-Lebedev Transformation on Distributions

A new proof of the representation theorem for the Kontorovich-Lebedev transformation on distributions is given.

On the theory of the Kontorovich-Lebedev transformation on distributions

A new proof of the representation theorem for the Kontorovich-Lebedev transformation on distributions is given.

A convolution connected with the Kontorovich-Lebedev transform

A convolution connected with the Kontorovich-Lebedev transform

The Kontorovich–Lebedev transformation on distributions of compact support and its inversion

1. During the past decade a variety of integral transformations have been extended to various classes of distributions. However, one important integral transformation, which has defied such generalization, is the Kontorovich–Lebedev transformation. The objective of this note is to alter that situation.

Parseval Relations for Kontorovich-Lebedev Transforms

The Kontorovich-Lebedev Transform of a function f(r), 0<r<∞, may be written in a general form as fg(μ) where,

An application of the Kontorovich-Lebedev Transform

In this note a simple application of the Kontorovich-Lebedev transform is described. The problem to which the transform is applied is the one considered in a recent paper (1).