A distribution associated with the Kontorovich-Lebedev transform

Semyon B. Yakubovich

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 \(\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)\), \(|\text{Re}\,z|\to\infty\), \(\alpha\gt 1\) uniformly in \(\overline{G_a}\). The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.

Keywords: Kontorovich-Lebedev transform, distributions, modified Bessel functions.

Mathematics Subject Classification: 46F12, 44A15, 33C10.

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