Opuscula Math. 26, no. 1 (2006), 161-172

Opuscula Mathematica

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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• Semyon B. Yakubovich
• University of Porto, Faculty of Sciences, Department of Pure Mathematics, Campo Alegre st., 687, 4169-007 Porto, Portugal