Browsing by Subject "Kontorovich-Lebedev transform"

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Access status: Open Access ,
A distribution associated with the Kontorovich-Lebedev transform
(2006) Yakubovich, Semyon B.
We show that in a sense of distributions $\lim_{\varepsilon\to 0+} {1\over \pi^2} \tau\sinh\pi\tau \int_{\varepsilon}^{\infty} K_{i\tau}(y)K_{ix}(y){dy\over y} =\delta(\tau-x),$ where $\delta$ is the Dirac distribution, $\tau#, $x\in\mathbb{R}$ and $K_{\nu}(x)$ is the modified Bessel function. The convergence is in $\mathcal{E}^{\prime}(\mathbb{R})$ for any even $\varphi(x)\in\mathcal{E}(\mathbb{R})$ being a restriction to $\mathbb{R}$ of a function $\varphi(z)$ analytic in a horizontal open strip $G_a=\{z\in\mathbb{C}\colon\,|\text{Im}\,z|\lt a, \ a\gt 0\}$ and continuous in the strip closure. Moreover, it satisfies the condition $\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)$, $|\text{Re}\,z|\to\infty$ uniformly in $\overline{G_a}$. The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.
Access status: Open Access ,
A double index transform with a product of Macdonald's functions revisited
(2009) Âkuboviĉ, Semën B.
We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions $K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)$, where $(x, y) \in \mathbb{R}_+ \times \mathbb{R}_+$ and $i \tau, \tau \in \mathbb{R}_+$ is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function $K_{i \tau}(t)$ as a kernel.
Access status: Open Access ,
A radial version of the Kontorovich-Lebedev transform in the unit ball
(2011) Âkuboviĉ, Semën B.; Vieira, Nelson
In this paper we introduce a radial version of the Kontorovich-Lebedev transform in the unit ball. Mapping properties and an inversion formula are proved in $L_p$.
Access status: Open Access ,
Beurling's theorems and inversion formulas for certain index transforms
(2009) Âkuboviĉ, Semën B.
The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Alacdoriald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform.