Kontorovich Lebedev Research Articles

The heat kernel associated with generalized Kontorovich–Lebedev transform

ABSTRACT We study some properties of the eigenfunction φ α , iτ μ , β of the following second-order differential operator defined on ( 0 , ∞ ) , by Δ x , α μ , β = x 2 d 2 d x 2 + ( 1 − 2 α ) x d d x − β 2 μ 2 x 2 μ ; α , β , μ ∈ ( 0 , ∞ ) . We obtain estimates of the integral of the kernel w α , β , μ , translation and convolution product related to φ α , iτ μ , β . We define and explore a generalization of the Kontorovich–Lebedev transform. The heat kernel and Weierstrass transform tied to Δ x , α μ , β are considered.

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

ABSTRACT The main objective of this work is to introduce the generalized convolution with trigonometric weighted γ = sin ⁡ y involving the Fourier cosine–sine and Kontorovich–Lebedev transforms, and to study its fundamental results. We establish the boundedness properties in a two-parametric family of Lebesgue spaces for this convolution operator. Norm estimation in the weighted L p space is obtained and applications of the corresponding class of convolution integro-differential equations are discussed. The conditions for the solvability of these equations in L 1 space are also founded.

The generalized Kontorovich–Lebedev transform and associated operators

The generalized Kontorovich–Lebedev transform and associated operators

Kontorovich–Lebedev wavelet transform on Besov type spaces

In the present paper, we define Besov type spaces associated with the Kontorovich–Lebedev transform. We widen the concept of continuous Kontorovich–Lebedev wavelet transform on space and derive continuity of Kontorovich–Lebedev wavelet transform on Besov type spaces and lastly characterize the Besov Kontorovich–Lebedev space by using Kontorovich–Lebedev wavelet coefficients.

Localization operators associated with the Kontorovich–Lebedev wavelet transform

Localization operators associated with the Kontorovich–Lebedev wavelet transform

Laplace transform-type multipliers for Kontorovich–Lebedev transform

In this paper, we investigate Poison kernel associated with Kontorovich–Lebedev transform. We significantly simplify the integral yielding a tractable form which is more useful for explicit calculation. We establish also that Kontorovich–Lebedev multipliers of Laplace transform type are bounded from [Formula: see text] into it self when [Formula: see text] and from [Formula: see text] into [Formula: see text] provided that [Formula: see text] is in the Muckenhoupt class [Formula: see text] on [Formula: see text]. At the end, an integral representation of the operator [Formula: see text] is obtained and its boundedness has been discussed in Lebesgue space.

Uncertainty principles for the continuous Kontorovich Lebedev wavelet transform

Uncertainty principles for the continuous Kontorovich Lebedev wavelet transform

Localization operators for the windowed Kontorovich Lebedev transform

Localization operators for the windowed Kontorovich Lebedev transform

Tikhonov regularisation for the Weierstrass transform associated with the Kontorovich–Lebedev transform

Using the best approximations and the theory of reproducing kernels, we will give real inversion formulas for the Weierstrass transform associated with the Kontorovich–Lebedev transform, and we will check the estimates of extremal functions.

Kuznetsov, Petersson and Weyl on GL(3), II: The generalized principal series forms

This paper develops the study by analytic methods of the generalized principal series Maass forms on GL(3). These forms occur as an infinite sequence of one-parameter families in the two-parameter spectrum of GL(3) Maass forms, analogous to the relationship between the holomorphic modular forms and the spherical Maass cusp forms on GL(2). We develop a Kuznetsov trace formula attached to these forms at each weight and use it to prove an arithmetically-weighted Weyl law, demonstrating the existence of forms which are not self-dual. Previously, the only full level, generalized principal series forms that were known to exist on GL(3) were the self-dual forms arising from symmetric-squares of GL(2) forms. The Kuznetsov formula developed here should take the place of the GL(2) Petersson trace formula for theorems “in the weight aspect”. As before, the construction involves evaluating the Archimedian local zeta integral for the Rankin–Selberg convolution and proving a form of Kontorovich–Lebedev inversion.

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 is the imaginary unit and is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when . The case reduces to the Kontorovich–Lebedev transform.

A pair of Kontorovich–Lebedev type transforms and associated pseudo-differential operators

A pair of Kontorovich–Lebedev type transforms and associated pseudo-differential operators

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.

On the orthogonality and convolution orthogonality via the Kontorovich–Lebedev transform

Notions of orthogonality and convolution orthogonality are explored with the use of the Kontorovich–Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral representations, orthogonality relations and explicit expressions are established.

Functional difference equations and eigenfunctions of a Schrödinger operator with δ' -interaction on a circular conical surface.

Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich-Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).

Boundedness of integral operators associated with the Kontorovich–Lebedev transform in the Lebesgue spaces type

In this paper, we prove the boundedness for the maximal and fractional maximal operators and Riesz potential-type operator associated with the Kontorovich–Lebedev transform (KL transform)in the [Formula: see text] spaces.

On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications

We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms $$ \left(h\underset{F_s,K}{\ast }f\right)(x) $$ in a two-parameter function space $$ {L}_p^{\alpha, \beta}\left({\mathrm{\mathbb{R}}}_{+}\right) $$ . We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution. We also impose necessary and sufficient conditions on the kernel h to ensure that the generalized convolution transform $$ {D}_h:f\mapsto {D}_h\left[f\right]=\left(1-\frac{d^2}{dx^2}\right)\left(h\underset{F_s,K}{\ast }f\right)(x) $$ is a unitary operator in L2(ℝ+) (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the Lp-norm.

On the boundedness of fractional maximal and Riesz potential operators associated with Kontorovich–Lebedev transform

ABSTRACT We define the maximal and fractional maximal operators associated with the Kontorovich–Lebedev transform. We study the boundedness of these operators in the space . Using these results we establish the Hardy–Littlewood–Sobobev theorem for the Riesz potential operator associated with Kontorovich–Lebedev transform.

$$L_{\alpha }^{p}$$-boundedness of the pseudo-differential operators associated with the Kontorovich–Lebedev transform

In this article, the authors prove a potentially useful $$L_{\alpha }^{p}$$-boundedness result for the pseudo-differential operators involving the Kontorovich–Lebedev transform. Several other related recent investigations, which dealt with the Kontorovich–Lebedev transform and the pseudo-differential operators are also considered for motivation purposes.

Analytical methods for perfect wedge diffraction: A review

The subject of diffraction of waves by sharp boundaries has been studied intensively for well over a century, initiated by groundbreaking mathematicians and physicists including Sommerfeld, Macdonald and Poincaré. The significance of such canonical diffraction models, and their analytical solutions, was recognised much more broadly thanks to Keller, who introduced a geometrical theory of diffraction (GTD) in the middle of the last century, and other important mathematicians such as Fock and Babich. This has led to a very wide variety of approaches to be developed in order to tackle such two and three dimensional diffraction problems, with the purpose of obtaining elegant and compact analytic solutions capable of easy numerical evaluation.The purpose of this review article is to showcase the disparate mathematical techniques that have been proposed. For ease of exposition, mathematical brevity, and for the broadest interest to the reader, all approaches are aimed at one canonical model, namely diffraction of a monochromatic scalar plane wave by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The first three approaches offered are those most commonly used today in diffraction theory, although not necessarily in the context of wedge diffraction. These are the Sommerfeld–Malyuzhinets method, the Wiener–Hopf technique, and the Kontorovich–Lebedev transform approach. Then follows three less well-known and somewhat novel methods, which would be of interest even to specialists in the field, namely the embedding method, a random walk approach, and the technique of functionally-invariant solutions.Having offered the exact solution of this problem in a variety of forms, a numerical comparison between the exact solution and several powerful approximations such as GTD is performed and critically assessed.