TY  - JOUR
ID  - Yakubovich2006
LB  - Yakubovich2006
AU  - Yakubovich, Semyon B.
TI  - A distribution associated with the Kontorovich-Lebedev transform
ST  - A distribution associated with the Kontorovich-Lebedev transform
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2006
VL  - 26
IS  - 1
SP  - 161
EP  - 172
DO  - http://dx.doi.org/10.7494/OpMath.2006.26.1.161
UR  - http://dx.doi.org/10.7494/OpMath.2006.26.1.161
AB  - 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.
KW  - Kontorovich-Lebedev transform
KW  - distributions
KW  - modified Bessel functions
SN  - 1232-9274
ER  -