Browsing by Subject "modified Bessel functions"

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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 ,
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.