A distribution associated with the Kontorovich-Lebedev transform

Abstract

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.